hep-th/0301164

UT-03-02

Supercoset CFT’s for String Theories

on Non-compact Special Holonomy Manifolds

Tohru Eguchi, Yuji Sugawara and Satoshi Yamaguchi

eguchi@hep-th.phys.s.u-tokyo.ac.jp , sugawara@hep-th.phys.s.u-tokyo.ac.jp ,

yamaguch@hep-th.phys.s.u-tokyo.ac.jp

Department of Physics, Faculty of Science,University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract

We study aspects of superstring vacua of non-compact special holonomy manifolds

with conical singularities constructed systematically using soluble N = 1 superconformal

field theories (SCFT’s). It is known that Einstein homogeneous spaces G/H generate

Ricci flat manifolds with special holonomies on their cones ' R+ × G/H, when they

are endowed with appropriate geometrical structures, namely, the Sasaki-Einstein, tri-

Sasakian, nearly Kahler, and weak G2 structures for SU(n), Sp(n), G2, and Spin(7)

holonomies, respectively. Motivated by this fact, we consider the string vacua of the

type: Rd−1,1 × (N = 1 Liouville) × (N = 1 supercoset CFT on G/H) where we use

the affine Lie algebras of G and H in order to capture the geometry associated to an

Einstein homogeneous space G/H. Remarkably, we find the same number of spacetime

and worldsheet SUSY’s in our “CFT cone” construction as expected from the analysis

of geometrical cones over G/H in many examples. We also present an analysis on the

possible Liouville potential terms (cosmological constant type operators) which provide

the marginal deformations resolving the conical singularities.

1 Introduction

String theories/M-theory on special holonomy manifolds developing conical singularities

are of great importance by several reasons: Firstly, they exhibit the non-perturbative

quantum effects, such as gauge symmetry enhancements, due to the appearance of light

solitonic states [1, 2, 3]. Secondly, they are expected to provide frameworks to discuss

the holographic dualities with the local or non-local interacting theories defined on the

asymptotic boundaries [4, 5], which naturally generalize the AdS/CFT -correspondence

[6].

It is known that an arbitrary (m − 1)-dimensional Einstein space Xm−1 possesses a

Ricci flat metric on its k-dimensional cone C(Xm−1) of the form

ds2 = dr2 + r2ds2Xm−1

, (1.1)

where r is the radial coordinate, and the special holonomies on C(Xm−1) originate from

the “weak special holonomies” on Xm−1 [7, 8, 9, 10]. To be more precise, the SU(n),

Sp(n), G2 and Spin(7) holonomies on the cone C(Xm−1) are in one to one correspondence

with the Sasaki-Einstein (m = 2n) [11], tri-Sasakian (m = 4n) [12, 13], nearly Kahler

(m = 7) [14] and weak G2 (m = 8) structures [15] on Xm−1, respectively as proved

in [16]1. (See the table 1.) This fact is very useful to systematically construct special

holonomy manifolds with conical singularities, because the Einstein homogeneous spaces

Xm−1 = G/H endowed with these geometrical structures are well understood since the

old days of Kaluza-Klein supergravity (SUGRA) [17, 18, 19, 20, 21, 22, 23] (and [24] for a

review) as well as from the mathematical literature mentioned above [11, 12, 13, 14, 15, 16]

and [25].

On the other hand, there also exists extensive literature on the worldsheet approaches

to these conical backgrounds in string theory. Early literature for the conifold and K3-

singularity is [26] and [27]. More recent studies are given in [28, 29, 30, 31] for the SU(n)-

holonomies, emphasizing the role of N = 2 Liouville theory [32] and the holographically

dual descriptions based on the (wrapped) NS5-brane geometry (see also [33]). All of these

cases possess the worldsheet N = 2 superconformal symmetry and have been discussed

in [34, 35, 36, 37] from the viewpoints of the “non-compact extensions” of Gepner models

[38]. There are also several related results [39, 40, 41, 42] from the spacetime view points,

but with the RR-flux at infinity, and also a work based on the Hybrid formalism [43] is

given in [44].

While these constructions in the case of the N = 2 supersymmetry have been rather

successful, it is difficult to construct string vacua on the conical backgrounds with Spin(7)

and G2 holonomies, which possess at most the N = 1 worldsheet SUSY. Partial attempts

1A good review for physicists on these mathematical concepts is found in the paper [8].

1

to construct the string vacua of Spin(7) and G2 holonomies with conical singularities

have been given in [45, 46]. General structure of string theory on manifolds with Spin(7)

and G2 was discussed in [47, 48] in particular from the point of view of the existence of

extended chiral algebras. Structure of these chiral algebras has also been discussed in

[49, 50, 51]. There are also several results [52] for the CFT constructions of compact G2

and Spin(7) manifolds based on the geometrical method of Joyce [53].

The main purpose of this paper is to give a systematic way of constructing special

holonomy manifolds with conical singularities based on the solvable N = 1 SCFT’s,

which may be regarded as a natural generalization of the construction in the N = 2

category mentioned above. Our strategy is quite simple: We formally replace an Einstein

homogeneous space X = G/H by an N = 1 supercoset CFT M = (G× SO(n)) /H, (n =

dimG − dimH) based on the affine Lie algebra of G and H. SO(n) stands for n free

fermions. We then add the N = 1 Liouville sector in place of the radial degrees of

freedom. We may call our construction as the “CFT cone” as opposed to the original

geometrical cone construction. Of course, one should keep in mind that the coset CFT

(WZW model) of G/H is not identical to the non-linear σ-model with the target manifold

G/H because of the presence of NS B field in WZW models. Nevertheless, as we will see

in the following, taking the supercoset CFT associated with the Einstein homogeneous

space provides a very good ansats for the superstring vacua of special holonomy. We

completely classify these coset constructions (at least for the cosets G/H with compact

simple groups G), which include the models found in [45, 46] as well as many of the vacua

in the N = 2 category presented in [26, 27, 28, 29, 30, 31]. Among other things, we will

find that our CFT cone approach leads to the right amount of worldsheet and spacetime

SUSY’s as expected from geometrical grounds in many examples.

This paper is organized as follows: In section 2 we clarify the properties of conformal

blocks characterizing the special holonomies in our CFT cone ansats. We also demonstrate

the explicit constructions of spacetime supercharges. In section 3, which is the main part

of this paper, we exhibit the classification of our string vacua with diverse spacetime

dimensions, and observe how we obtain the special holonomies which coincides (or differs)

with those of the geometrical constructions of Ricci flat cones. In section 4 we analyse

the spectrum of possible cosmological constant type operators, which will provide the

marginal deformations resolving the conical singularities in backgrounds. Section 5 is

devoted to discussions on open problems.

2

dim name holonomy Killing spinor ] spacetime SUSY base of the cone

4 hyper Kahler SU(2) (2,0) 16 tri-Sasakian

6 Calabi-Yau SU(3) (1,1) 8 Sasaki-Einstein

7 G2 G2 1 4 nearly Kahler

8 hyper Kahler Sp(2) (3,0) 6 tri-Sasakian

8 Calabi-Yau SU(4) (2,0) 4 Sasaki-Einstein

8 Spin(7) Spin(7) (1,0) 2 weak G2

Table 1: We summarize the relation between the special holonomies on the cone C(Xm−1)

and the geometrical structures on the Einstein spaces Xm−1.

2 Superstring Vacua of Non-compact Special Holon-

omy Manifolds as the ‘Cone over SCFT’s’

2.1 General Set Up and Aspects of Spacetime SUSY

In this paper we shall search for supersymmetric string vacua with the form

Rd−1,1 × Rφ ×M , (2.1)

where Rφ denotes the N = 1 Liouville theory (N = 1 linear dilaton SCFT) with the

background charge Qφ and M is a rational N = 1 SCFT with a central charge cM.

We would like to identify this conformal system as a string vacuum of singular special

holonomy manifold realized as the “cone over M” in the decoupling limit gs → 0. By

assumption we have a linear dilaton background Φ(φ) = −Qφ

2φ. The weak coupling region

φ ∼ +∞ corresponds to the holographic boundary Rd−1,1 and the opposite region φ ∼ −∞is located around the singularity (“tip of the cone”).

The criticality condition is written as

3

2(d− 2) +

(3

2+ 3Q2

φ

)+ cM = 12 , (2.2)

or equivalently,

Q2φ

8=

9− d

16− cM

24. (2.3)

To build up the consistent string vacua the modular invariance is an important crite-

rion. Let us first discuss the values of Liouville momentum which enters into the modular

invariant partition function. We recall that the conformal dimension of a Liouville expo-

nential eγφ is given by

h(eγφ) = −12γ2 − 12Qφγ = −12(γ +Qφ2)2 +Q2φ8 (2.4)

3

It is known that the Liouville momentum in the partition function takes values in the

“principal series” (at least in the semi-classical analysis) [54]

γ = −Qφ2 + ip, p ∈ R (2.5)

and thus

h(eγφ) = 12p2 +Q2φ8 ≥ Q2

φ8. (2.6)

Liouville exponential with momentum in the principal range represents a plane wave

propagating in spacetime and thus is a delta-function normalizable state. We note the

presence of a “mass” gap Q2φ/8 in the Liouville spectrum. Therefore, conformal blocks are

expanded only in terms of the “massive representations” of the extended superconformal

algebra which characterizes each special holonomy. The list of massive representations

are given in appendix B.

We here summarize the aspects of spacetime SUSY in (2.1) for various spacetime

dimensions for later convenience.

• SU(n)-holonomy :

This case corresponds to d = 10 − 2n, and the worldsheet SUSY is required to be

enhanced to N = 2. More explicitly, the criterion for the SU(n)-holonomy is to ask

whether it is possible to rewrite the theory as2

Rφ ×M ∼= [Rφ × U(1)]× MU(1)

, (2.7)

where Rφ × U(1) stands for the N = 2 Liouville theory and the coset M/U(1)

describes an N = 2 SCFT. In the case when M is an N = 1 coset, M/U(1) should

be a Kazama-Suzuki model [56] associated to a Kahler homogeneous space.

When making use of this criterion (2.7), the most crucial point is as follows: The

N = 2 Liouville theory includes a compact boson Y with the radius equal to Qφ

whose conformal blocks are written in terms of the theta functions of the level

n2 · 2/Q2φ, where n is an integer, as discussed in [34, 36]. Ambiguity n2 of the level

2In this paper we often use the concise expressions such as

M∼= MHk

×Hk ,

where Hk denotes the conformal theory of the level k H-current algebra. Precisely speaking, if H

is abelian, the R.H.S must be interpreted as an orbifoldization with respect to the H-charge as in the

Gepner model [38]. If H is semi-simple, the R.H.S strictly means the “projected tensor product” discussed

in [55].

4

originates from the choice of possible momentum lattice of compact boson as shown

by the theta function identity (A.6).

We have a natural geometric interpretation of (2.7): The Calabi-Yau cones are

build up over Sasaki-Einstein spaces, which have the U(1) fibration over the Kahler-

Einstein manifolds

M U(1)−→ M/U(1) . (2.8)

The standard GSO projection with respect to the total U(1)R-charge of N = 2

SCA leads to the spacetime SUSY as in Gepner models. The explicit construction

of supercharges is given in [29] and we can check the cancellation of conformal blocks

directly.

The characteristic features of the conformal blocks for the SU(n)-holonomy in each

case n = 2, 3, 4 is summarized as follows;

(i) SU(2)-holonomy : In this case the conformal blocks are expanded by the

massive characters ofN = 4 superconformal algebra with c = 6 (level 1), which

have the forms as qh−∗θ23/η

3 in the NS sector [59]. The SUSY cancellation is

expressed by the familiar Jacobi identity(θ3η

)4

−(θ4η

)4

−(θ2η

)4

≡ 0 , (2.9)

and we have 16 unbroken supercharges in 6-dimensional spacetime.

(ii) SU(3)-holonomy : In this case the conformal blocks are expanded by the

massive characters of “c = 9 extended superconformal algebra” [60, 61], which

have the form qh−∗Θ∗,1θ3/η3 in the NS sector [60]. The SUSY cancellation is

expressed as the following identities[(θ3η

)2

−(θ4η

)2]

Θ0,1

η−

(θ2η

)2Θ1,1

η≡ 0 ,

[(θ3η

)2

+

(θ4η

)2]

Θ1,1

η−

(θ2η

)2Θ0,1

η≡ 0 , (2.10)

and we have 8 supercharges in 4-dimensional spacetime.

(iii) SU(4)-holonomy : In this case the conformal blocks are expanded by the

massive characters of “c = 12 extended superconformal algebra”, which take

the form qh−∗Θ∗,3/2θ3/η3 in the NS sector [62]. The SUSY cancellation is

5

ensured by the identities

θ3η

Θ0, 32

η(τ)− θ4

η

Θ0, 32

η(τ)− θ2

η

Θ 32, 32

η(τ) ≡ 0,

θ3η

Θ1, 32

η(τ) +

θ4η

Θ1, 32

η(τ)− θ2

η

Θ 12, 32

η(τ) ≡ 0 , (2.11)

and we have 4 supercharges in 2-dimensional spacetime.

• Sp(n)-holonomy :

These superstring vacua correspond to d = 10− 4n and are described by the N = 4

SCFT of c = 6n (level n). In the case when M is defined as an N = 1 supercoset

CFT, the condition for the N = 4 enhancement of worldsheet SUSY has been

studied in [57]. We have the (small) N = 4 worldsheet SUSY, if and only if the

following rewriting is possible;

Rφ ×M ∼= [Rφ × SU(2)]× MSU(2)

, (2.12)

where the N = 1 coset M/SU(2) is associated to a Wolf space (quaternionic sym-

metric space) [58].

In the similar manner as (2.8), the tri-Sasakian homogeneous spaces are known to

have SU(2)-fibrations over Wolf spaces;

M SU(2)−→ M/SU(2) , (2.13)

which yields the geometrical interpretation of (2.12).

Concerning the structure of conformal blocks, the case of Sp(1)(∼= SU(2))-holonomy

was already illustrated above. In the case of Sp(2)-holonomy, on the other hand,

the conformal blocks should be expanded by the massive characters of N = 4 SCA

with c = 12, which are written [59] as

qh−∗

η

(θ3η

)2

χSU(2)1∗ ≡ qh−∗

η

(θ3η

)2Θ∗,1η

. (2.14)

The SUSY cancellation is expressed by the identity (2.10) again. However, it turns

out that we obtain the N = (3, 3) (or N = (6, 0)) spacetime SUSY, that is, there

exist only 6 supercharges in 2-dimensional spacetime.

6

• G2-holonomy :

We set d = 3, and the spacetime SUSY requires the condition

U(1)3/2 ⊂M , (2.15)

where U(1)3/2 denotes a c = 1 CFT whose conformal blocks are described by char-

acters Θ∗,3/2(τ)/η(τ). Precisely this relation means that the conformal blocks of

the M-sector are expanded in terms of characters Θ∗,3/2(τ)/η(τ) and the branching

coefficients are not hit by the GSO projection. The SUSY cancellation is realized

again as (2.11). If one takes account of the Liouville fermion, we have a relation [51]

SO(1)1 × U(1)3/2∼= SO(6)1

SU(3)1× SO(1)1

∼= SO(7)1

(G2)1× (G2)1

SU(3)1

∼= tri-critical Ising× 3-state Potts , (2.16)

Thus our condition is consistent with the criterion for the G2-holonomy given in

[47].

• Spin(7)-holonomy :

We set d = 2, and the existence of spacetime SUSY requires the condition

tri-critical Ising ⊂M , (2.17)

which precisely means that the conformal blocks in theM-sector can be decomposed

by the N = 1 characters of the tri-critical Ising model (the m = 3 model in the

N = 1 minimal series with c = 7/10). The SUSY cancellation is realized as the

7

following identities 3; √θ3ηχtri NS

0 −√θ4ηχtri NS

0 −√θ22ηχtri R

7/16 ≡ 0

√θ3ηχtri NS

1/10 +

√θ4ηχtri NS

1/10 −√θ22ηχtri R

3/80 ≡ 0 . (2.18)

dim name worldsheet SUSY algebra structure of massive reps.

4 hyper Kahler N = 4 SU(2)1 SO(4)1

6 Calabi-Yau N = 2 U(1)3/2 SO(2)1 × SU(2)1

7 G2 N = 1 tri-critical Ising SO(1)1 × U(1)3/2

8 hyper Kahler N = 4 SU(2)2 SO(4)1 × SU(2)1

8 Calabi-Yau N = 2 U(1)2 SO(2)1 × U(1)3/2

8 Spin(7) N = 1 Ising SO(1)1× tri-critical Ising

Table 2: We summarize the algebraic structures of worldsheet theories describing man-

ifolds with special holonomies. The “structure for the massive reps.” is the algebraic

structure which is manifest in the characters of massive representations.

2.2 Constructions of Spacetime Supercharges

Although we have just seen the characteristic features of conformal blocks in the string

vacua (2.1), it may still be helpful to construct explicitly the spacetime supercharges for

these vacua. We shall only consider the left-movers for simplicity. It is straightforward to

combine the left and right movers so as to be consistent with the GSO conditions for the

type IIA or type IIB string theories.

3In the convention here, the N = 1 characters of tri-critical Ising are written in terms of the N = 0

ones as follows;

χtriNS0 = χtri

0 + χtri3/2 , χtri NS

0 = χtri0 − χtri

3/2 , χtriNS1/10 = χtri

1/10 + χtri3/5 , χtri NS

1/10 = χtri1/10 − χtri

3/5 ,

χtri R7/16 = 2χtri

7/16 , χtri R3/80 = 2χtri

3/80 .

8

1. Supercharges for the SU(n)-holonomies :

Construction of supercharges in these cases is quite standard and presented in [29]. We

prepare the spin fields Sα (±)0 for the Minkowski spacetime Rd−1,1 with the conformal weight

h = d/16, where α = 1, 2, . . . , 2[d−22 ], and ± denotes the chirality. We also introduce the

spin fields for the sector of internal space described by the N = 2 SCFT

Rφ ×M ∼= [Rφ × U(1)]× MU(1)

. (2.19)

The total U(1)R current JR is bosonized as

JR = i∂H , H(z)H(0) ∼ −n ln z , (2.20)

and the relevant spin fields are defined as

S± = e±i2H , (2.21)

which have the conformal weight h = n/8.

The GSO projection gives the chiral supercharges for d = 2, 6, and non-chiral super-

charges for d = 4;

d = 2, 6 : Q±α =

∮S

α (+)0 S±e−

ϕ2 , (2.22)

d = 4 : Q+α =

∮S

α (+)0 S+e−

ϕ2 , Q−α =

∮S

α (−)0 S−e−

ϕ2 , (2.23)

where ϕ denotes the standard bosonized superghost.

2. Supercharges for the Spin(7)-holonomy

The Spin(7)-holonomy admits one Killing spinor with a definite chirality. We thus

assume the spin-field in R1,1 should have a definite chirality, say, S(+)0 . 4 In order to define

the spin fields for the internal space, we treat the Liouville andM sectors separately. Since

the N = 1 Liouville sector includes a free fermion, we have the doubly degenerate spin

fields σφ±, which have the conformal weight h = 1/16 and satisfy the OPEs

ψφ(z)σφ±(0) ∼ ±i√

2z1/2σφ∓ , σφ

±(z)σφ±(0) ∼ 1

z1/8, σφ

+(z)σφ−(0) ∼ z3/8

√2ψφ(0).(2.24)

4Without this assumption, we would have twice as many supercharges as those expected

from supergravity. In fact, we further construct the BRST invariant supercharge Q =∮S

(−)0

(σφ

+σM− + σφ−σM+

)e−ϕ/2 (Q′ =

∮S

(−)0

(σφ

+σM+ + σφ−σM−

)e−ϕ/2) which is mutually local with

Q (Q′) (2.26). This disagreement is probably an artifact originating from the fact that we are now work-

ing on the singular background without Liouville potential terms, while the 2-dimensional supergravity

should correspond to low energy effective theory on smooth backgrounds.

9

On the other hand, the assumption (2.17) implies the existence of spin fields from tri-

critical Ising model with h = 7/16. They are again doubly degenerate and we denote

them as σM± .

The candidate supercharges are now written as ∼∮S

(+)0 σφ

∗σM∗ e−ϕ/2. To achieve

the correct construction we must take account of the BRST invariance, especially the

condition G0(≡ Gφ0 +GM0 ) = 0. The following relation is helpful for this purpose;

GM(z)σM± (0) ∼ 1

z3/2

√7

16− cM

24σM∓ (0)

=1

z3/2

Qφ

2√

2σM∓ (0) , (2.25)

where the first line is deduced from (GM0 )2 = LM0 − cM24

, and the second line is due to

(2.3). Now, it is not difficult to find out the following BRST invariant combinations;

Q =

∮S

(+)0

(σφ

+σM+ + σφ

−σM−

)e−ϕ/2 , Q′ =

∮S

(+)0

(σφ

+σM− + σφ

−σM+

)e−ϕ/2 . (2.26)

They are mutually non-local and we must take only one of them, which amounts to process

of GSO projection. We have thus obtained the desired supercharge.

3. Supercharges for the G2-holonomy

Construction of supercharges for the G2-holonomy is almost parallel. The condition

(2.15) ensures the existence of the doubly degenerate spin fields σM± with h = 3/8, and

the BRST invariant supercharges are found out to be

Qα =

∮Sα

0

(σφ

+σM+ + σφ

−σM−

)e−ϕ/2 , Q

′ α =

∮Sα

0

(σφ

+σM− + σφ

−σM+

)e−ϕ/2 , (2.27)

where Sα0 (α = 1, 2) is the 3-dimensional spin field with the conformal weight h = 3/16.

Qα and Q′ α are again mutually non-local, and we have to take either one by the GSO

projection.

4. Supercharges for the Sp(2)-holonomy

The last case is the most non-trivial. The Sp(2)-holonomy admits three independent

Killing spinors with the same chirality. So, we assume the longitudinal part of spin fields

is S(+)0 , as in the case of Spin(7)-holonomy.

To define the spin fields in the internal space, we first recall the SU(2)2-current algebra

{Ka(z)} describing the SU(2)R-symmetry in the N = 4 superconformal theory, and the

total U(1)R current of the N = 2 SCFT is identified as 2K3(z). Moreover, since all

conformal blocks are expanded into massive characters (2.14), the SU(2)2 current algebra

10

is also embedded in SU(2)1×SO(4)1. In particular, we have the following decomposition

of the total U(1)R-current;

2K3(z) = i∂H1(z) + i∂H2(z) +√

2i∂H3(z) , (2.28)

Hi(z)Hj(0) ∼ −δij ln z . (2.29)

Here the compact bosons H1, H2 have radii equal to 1, and describe 1+3=4 free fermions

corresponding to the superpartners of the Liouville mode and the SU(2)-factor in (2.12)

(the fermions along the fiber of (2.13)). H3 has the radius equal to√

2 (self-dual radius)

and generates the SU(2)1-current algebra. We now introduce the following spin fields (up

to cocycle factors)

Sε1ε2ε3 = eiε12

H1+iε22

H2+i√

2ε32

H3 , (εi = ±1) . (2.30)

Candidate supercharges are now (linear combinations of)

∮S

(+)0 Sε1ε2ε3e−ϕ/2 which have

8 independent components. Again the non-trivial condition for the BRST invariance is

the constraint G0 ≡ GM0 +Gφ0 = 0, and one finds the following solutions

Q± =

∮S

(+)0 S±±±e−ϕ/2 ,

Q3 =1

2

∮S

(+)0

(S++− + S−−+

)e−ϕ/2 , (2.31)

and also

Q′ =1

2

∮S

(+)0

(S+−+ − S−+−)

e−ϕ/2 . (2.32)

As is easily shown, Q+, Q− are the same supercharges as those of SU(4)-holonomy (2.22).

Moreover, Qa(a = ±, 3) composes a triplet of SU(2)R and Q′ is a singlet. In particular,

{Qa} are identified as the spin 1 primary fields of the current algebra Ka(z)(a = ±, 3). Qa

and Q′ are mutually non-local, and we should choose either one by the GSO projection.

We take the triplet Qa to recover the string vacua of Sp(2)-holonomy.

Finally let us make a comment. It is well-known [7, 8, 9, 10] that the Ricci flat

background Rd−1,1 × C(X9−d) is converted into AdSd+1 ×X9−d by letting (d− 1)-branes

“fill” the Minkowski spacetime Rd−1,1, and taking the near horizon limit. We here denote

the (9− d)-dim. Einstein space as X9−d and its Ricci flat cone as C(X9−d). Even though

the worldsheet approach to string theory on such backgrounds is usually very difficult due

to the RR-flux, the d = 2 cases are tractable. Namely, we can fill the NS1 branes and

interpolate the background R1,1 × C(X7) to AdS3 ×X7.

11

The analogous relation in our study of “cone over SCFT” could be depicted as

R1,1 × Rφ ×M +NS1

=⇒ AdS3 ×M . (2.33)

In the R.H.S, the AdS3 sector is described by the SL(2; R) super WZW model with the

(bosonic) level k which is determined by the relation Q2φ = 2

k+2. This relation (2.33) has

been already pointed out in [29] in the cases of SU(4)-holonomy. There it is also discussed

that the spacetime SUSY algebra should be enhanced to the N = 2 superconformal

algebra (only for the left or right movers) acting on the boundary of AdS3, which is

explicitly constructed in [63] along the same line as [64]. The similar observations are also

possible for the Spin(7) and Sp(2)-holonomy cases:

(i) Spin(7)-holonomy : The easiest way to move from the L.H.S to R.H.S in (2.33) is

by making the formal replacements; ψ0 → iΨ2, ψ1 → Ψ1, and ψφ → iΨ3, where

ψ0, ψ1, ψφ are the free fermions along the longitudinal and Liouville directions

and ΨA denotes the fermionic coordinates in the N = 1 SL(2; R)-WZW model.

The structure of spin fields are quite similar. The essential difference lies in the

BRST charges, especially in the definitions of superconformal currents G(z): G(z)

of SL(2; R) super WZW model includes a cubic fermionic term ∼ Ψ1Ψ2Ψ3, while

that of N = 1 Liouville theory (×R1,1) only includes a linear term ∼ Qφi∂ψφ.

With these preparations, we can construct twice as many supercharges for the

AdS3 ×M backgrounds as compared with (2.26);

G+1/2 =

∮S

(+)0

(σ3

+σM+ + σ3

−σM−

)e−ϕ/2 ,

G−1/2 =

∮S

(−)0

(σ3

+σM− − σ3

−σM+

)e−ϕ/2 , (2.34)

where we denote the spin fields associated Ψ1, Ψ2 as S(+)0 , and σ3

± is defined similarly

as in (2.24) with respect to Ψ3. They are naturally identified as the “zero-modes”

of the spacetime N = 1 superconformal algebra. It is also not difficult to con-

struct the full superconformal current oscillators based on the Wakimoto free field

representation along the line of [64].

(ii) Sp(2)-holonomy : For the Sp(2)-holonomy the argument is almost parallel. In the

12

background AdS3 ×M we can obtain twice as many supercharges

G±+1/2 =

∮S

(+)0 S±±±e−ϕ/2 ,

G3+1/2 =

1

2

∮S

(+)0

(S++− + S−−+

)e−ϕ/2 ,

G±−1/2 =

∮S

(−)0 S∓±±e−ϕ/2 ,

G3−1/2 =

1

2

∮S

(−)0

(S−+− + S+−+

)e−ϕ/2 . (2.35)

They are again enhanced to the full generators of N = 3 superconformal algebra

[65], where the superconformal currents behave as a triplet of SU(2)R symmetry,

and coincide with those constructed in [66] (up to normalizations and the convention

of spin fields). In addition, we will show in the next section that the Sp(2)-holonomy

can be achieved only for M = SO(5)/SO(3), SU(3)/U(1), if we assume supercoset

CFT’s for the M-sector. This fact is consistent with the observation given in [66].

3 Supercoset CFT’s for Superstring Vacua of Special

Holonomies

3.1 Preliminary: Some Notes on Supercoset CFT’s

In order to fix our discussions we shall adopt the coset construction for the M-sector

from now on. The N = 1 supercoset CFT’s are easily constructed by using the super

Kac-Moody algebras. They have the general form

M =Gk × SO(D)1

H, dimG/H = D , (3.1)

with a Lie group G and its subgroup H . SO(D)1 stands for the current algebra generated

by D free fermions. The condition dimG/H = D ensures the N = 1 superconformal

symmetry. We especially concentrate on the cases of D = 9− d for the superstring vacua

of the type Rd−1,1 × Rφ ×M (namely, for the 10 − d dimensional internal space). The

reason why we do so is as follows: The coset spaces G/H are generically endowed with

Einstein metrics and were studied extensively in old days of Kaluza-Klein supergravity

[17, 18, 19, 20, 21, 22, 23, 24]. The cones over these spaces are known to possess Ricci

flat metrics [7, 8, 9, 10], leading to supersymmetric string vacua, when the coset spaces

13

possess the geometrical structures listed in the table 1. In a formal sense the string vacua

(2.1) we are studying are the CFT versions of the Ricci flat cones. We point out that

cM ≤ 3D/2 holds in general (equality holds at k = ∞). Thus we always have the real

background charge Qφ. (Recall the condition (2.3).)

The level of the current algebra of H is slightly non-trivial to determine and depends

on the embedding of H in general. We assume the decomposition H = H0×H1×· · ·×Hr,

where H0 is the abelian part and Hi (i = 1, . . . , r) are simple parts. The level ki of each

simple factor Hi is defined in the standard manner;

Ja(i)(z)J

b(i)(0) ∼ ki(t

a(i), t

b(i))i

z2+if ab

(i) c

zJc

(i)(0) (for i 6= 0) ,

Ja(0)(z)J

b(0)(0) ∼ k0(t

a(0), t

b(0))

z2(for H0) . (3.2)

where {ta(i)} is a basis of the Lie algebra of Hi and Ja(i)(z) ≡ (J(i)(z), t

a(i))i. The Killing

metrics ( , ) for G and ( , )i for the Hi sectors are canonically normalized as (θ, θ) = 2,

(θi, θi)i = 2, where θ, θi are the highest roots of G, Hi, respectively. Notice that the

inner products ( , ) and ( , )i have different normalizations in general. (θi, θi) is not

necessarily equal to 2 and depends on the choice of embedding of Hi. Now, the levels ki

are determined by comparing the Schwinger terms of the super affine Lie algebras of G

and Hi;

k0 = k + g∗

ki =2

(θi, θi)(k + g∗)− h∗i , (i = 1, . . . , r) , (3.3)

where g∗, h∗i denote the dual Coxeter numbers of G, Hi respectively5. Care is needed

when G is non-simply laced and some Hi’s are embedded along its short roots. The

central charges of these coset CFT’s are evaluated as

cM =k dimG

k + g∗+

1

2D −

r∑i=0

ki dimHi

ki + h∗i. (3.4)

We also remark that the sub-root lattice describing the charge spectrum of H0-sector

must be specified in order to define the coset model uniquely. The conformal blocks of

this sector are written in terms of the theta functions associated to the charge lattice

5In the case Hi = SO(3), one must use the formula

ki =1

(θi, θi)(k + g∗)− 1

instead of (3.3). This fact is due to the equivalence SO(3)k∼= SU(2)2k as an affine Lie algebra.

14

Γ ⊂ √k + g∗Q, where Q is the root lattice of G. They are explicitly written as

FH0λ (τ) =

Θ(Γ)λ (τ)

η(τ)L, (dimH0 = L)

Θ(Γ)λ (τ) =

∑α∈λ+Γ

q12(α,α) , (λ ∈ Γ∗) . (3.5)

For example, suppose that the charge lattice is given by

Γ = Zν1 + · · ·+ ZνL , (L = dimH0) ,

(νa, νb) = 0 , (∀a 6= b) , (νa, νa) = 2la(k + g∗) , (3.6)

then we have the decomposition of the theta functions

Θ(Γ)λ (τ) =

L∏a=1

Θma,la(k+g∗)(τ) , λ =

L∑a=1

maν∗a , (3.7)

where ν∗a are the dual bases of Γ∗ such that (νa, ν∗b ) = δab. In this paper we shall adopt the

convention that “U(1)k” means the c = 1 conformal theory composed of the conformal

blocks of the forms Θ∗,k(τ)/η(τ). Namely, we here find

(H0)k+g∗ ∼= U(1)l1(k+g∗) × · · · × U(1)lL(k+g∗) (3.8)

We will later face non-trivial examples in which the different choice of charge lattice leads

to inequivalent string vacua.

3.2 Coset Constructions of d = 6 Superstring Vacua

We start with the study of d = 6 string vacua. The spacetime SUSY corresponds to the

SU(2)-holonomy in this case. Assuming the N = 1 supercoset CFT M = G/H with

the condition that G is compact, simple and dimG/H = 3, the possible examples are

G = SU(2), H = {id}, and G = SO(4), H = SO(3).

The first example, G = SU(2), H = {id}, is the familiar CHS σ-model describing the

NS 5-brane [67];

Rφ × ψφ ×M = Rφ × ψφ × SU(2)k × SO(3)1

∼= [Rφ × ψφ × U(1)k+2 × SO(1)1

]× SU(2)k × SO(2)1

U(1)k+2. (3.9)

Liouville field, together with the WZW model SU(2)k, describes the configuration of the

throat region R+ × S3 transverse to the NS 5-brane. In the last line the part described

15

by [· · ·] is the N = 2 Liouville theory with Qφ =√

2/(k + 2) and the remaining part is

the N = 2 minimal model of the level k. In [27] this system is interpreted as describing

the string theory compactified on an ALE space with the Ak+1-type singularity. (The D

and E-type singularities are naturally incorporated as the modular data of SU(2)k sector

[28].) Thus (3.9) represents the well-known T-duality between NS 5-brane and ALE space

[27, 68].

In the second example, G = SO(4), H = SO(3), we have two possibilities of SO(3)-

embedding. The first choice is the embedding into one of the SO(3)’s of SO(4) ' SO(3)×SO(3), which again reduces to the CHS σ-model because of the relation

M =SO(4)k × SO(3)1

SO(3)k/2

∼= SU(2)k × SO(3)1 . (3.10)

The second choice is the diagonal embedding in SO(3)× SO(3), which leads to

M =SO(4)k × SO(3)1

SO(3)k+1

. (3.11)

This case corresponds to vacua with no spacetime SUSY.

In addition, we can consider the following generalization of (3.11)

M =SU(2)k1 × SU(2)k2 × SO(3)1

SU(2)k1+k2+2

. (3.12)

These N = 1 diagonal coset theories have been studied intensively in [50]. It is obvious

that (3.12) cannot provide any d = 6 supersymmetric vacuum. However, we will later

show that the d = 3 and d = 2 supersymmetric vacua of this form exist under the suitable

restrictions of the levels of current algebras.

3.3 Coset Constructions of d = 4 Superstring Vacua

In this case the supersymmetric string vacua correspond to SU(3)-holonomy. The criterion

for spacetime SUSY is thus whether the worldsheet SUSY is enhanced to N = 2.

We assume dimG/H = 5. If G is simple, we only have the possibilities

G/H = SO(6)/SO(5) , SU(3)/SU(2) . (3.13)

We also study the case G/H = (SU(2) × SU(2))/U(1) as a particular example of a

semi-simple G.

1. G/H = SO(6)/SO(5)

16

In this case the N = 1 supercoset CFT is given as

M =SO(6)k × SO(5)1

SO(5)k+1. (3.14)

The worldsheet SUSY for the total system Rφ × ψφ ×M is not enhanced since the coset

M/U(1) is not well-defined in this case. Therefore, this model corresponds to a string

vacuum with no spacetime SUSY.

2. G/H = SU(3)/SU(2)

We have two possibilities of the embedding of SU(2).

(i) SU(2) embedded as the “isospin subgroup” :

The easier embedding is of course into the usual isospin subgroup of SU(3). In this

case, we can show that the worldsheet SUSY is enhanced to N = 2 because of the

equivalence

Rφ × ψφ ×M ∼= [Rφ × ψφ × U(1)3(k+3) × SO(1)1]× SU(3)k × SO(4)1

SU(2)k+1 × U(1)3(k+3)

,

(3.15)

as we already mentioned in (2.7). The part [· · ·] is interpreted as the N = 2

Liouville. In fact, the criticality condition (2.3) gives us Q2φ = 6/(k+3), resulting in

32 · 2/Q2φ = 3(k + 3). Thus, the U(1)3(k+3) piece is identified as the compact boson

of N = 2 Liouville theory. The remaining coset CFT is the Kazama-Suzuki model

for CP2. In this way we have confirmed that this string vacuum corresponds to a

non-compact CY3 manifold.

Notice that the coset SU(3)/SU(2) is isomorphic with S5, which is an elementary

example of Sasaki-Einstein manifold with the U(1)-fibration

S5 U(1)−→ CP2 . (3.16)

This is the geometrical interpretation of (3.15).

(ii) SU(2) embedded as the maximal subgroup :

Another embedding is defined by identifying the simple root of SU(2) with θ/2,

where θ ≡ α1 + α2 is the highest root of SU(3). In practice, this is given by

restricting the canonical action of SU(3) on complex 3-vectors to that of SO(3)('SU(2)) on real 3-vectors. By this embedding, the adjoint representation of SU(3) is

decomposed as 8 → 3 + 5, and thus it is “maximal” (see [22] for the detail). The

coset space is again isomorphic with S5.

17

Since (θ/2)2 = 1/2 holds, we find that the relevant supercoset CFT should have the

following form due to the formula (3.3)

M =SU(3)k × SO(5)1

SU(2)4k+10

. (3.17)

It may be helpful to confirm that this coset CFT is really well-defined. To this

aim it will be enough to check the existence of SO(5)1/SU(2)10. In fact, this coset

CFT corresponds to the maximal embedding SU(2) ⊂ SO(5) which we later con-

sider. Another explanation is given as follows: We have a conformal embedding

SU(2)10 ⊂ SO(5)1 (c = 5/2 for both of SO(5)1, SU(2)10) based on the E6-type

modular invariance of SU(2)10 [70]. Namely, the corresponding partition function

is given by

Z =∣∣∣χ(10)

0 + χ(10)6

∣∣∣2 +∣∣∣χ(10)

3 + χ(10)7

∣∣∣2 +∣∣∣χ(10)

4 + χ(10)10

∣∣∣2 , (3.18)

where we denote the SU(2)k character of spin `/2 as χ(k)` , and the following character

relations are known (see [69], for example);

χSO(5)1basic (τ) = χ

(10)0 (τ) + χ

(10)6 (τ) , χ

SO(5)1spinor (τ) = χ

(10)3 (τ) + χ

(10)7 (τ) ,

χSO(5)1vector(τ) = χ

(10)4 (τ) + χ

(10)10 (τ) . (3.19)

We have no extra U(1) symmetry in (3.17) since the SU(2) is a maximal subgroup,

and thus the worldsheet SUSY cannot be enhanced. Therefore, the corresponding

string vacua are not supersymmetric.

3. G/H = (SU(2)× SU(2))/U(1)

Lastly, let us consider the most non-trivial example

M =SU(2)k1 × SU(2)k2 × SO(5)1

U(1). (3.20)

As was already illustrated, the charge lattice of U(1) must have the following form because

of the worldsheet SUSY

Γ = Z

(p√k1 + 2α+ q

√k2 + 2β

), (3.21)

where α, β denote the simple roots of the two SU(2) factors and p, q ∈ Z≥0 ((p, q) 6= (0, 0)).

18

We can reexpress M as

Rφ × ψφ ×M ∼= Rφ × ψφ × SU(2)k1 × SO(2)1

U(1)k1+2× SU(2)k2 × SO(2)1

U(1)k2+2

×U(1)k1+2 × U(1)k2+2

U(1)p2(k1+2)+q2(k2+2)

× SO(1)1

∼= [Rφ × ψφ × U(1)(k1+2)(k2+2){p2(k1+2)+q2(k2+2)} × SO(1)1]×Mk1 ×Mk2 ,

(3.22)

where Mk denotes the N = 2 minimal model with level k (c = 3kk+2

). In the last line,

we made use of the product formula of theta functions. The N = 2 worldsheet SUSY

requires that the part [· · ·] becomes the N = 2 Liouville theory. The criticality condition

(2.3) gives us

Q2φ =

2(k1 + k2 + 4)

(k1 + 2)(k2 + 2), (3.23)

and hence if and only if p = q holds, the correct factor U(1)(k1+2)(k2+2)(k1+k2+4) is obtained.

In conclusion, only in the cases p = q, we have the N = 2 worldsheet SUSY, leading to

the SU(3)-holonomy.

This type string vacua were studied in [36], and especially the simple cases of k1 = k,

k2 = 0 correspond to the CY3 singularity of Ak+1-type [29, 34, 71] (D, E-types are also

described by the modular data).

We also make a comment on a geometric interpretation. The Einstein homogeneous

space T p,q = (SU(2)× SU(2))/U(1) is defined by the U(1)-action

eiθ 7−→ (eipθσ32 , eiqθ

σ32 ) , (3.24)

with relatively prime integers p, q. It is well-known that only the case p = q = 1 allows

the spacetime SUSY [23]. More precisely, T p,q becomes a Sasaki-Einstein space only for

p = q = 1. The cone over T 1,1 is the well-known conifold. The parameters p, q precisely

correspond to those introduced in our discussions (at least for the cases k1 = k2). Thus

the condition for the presence of spacetime SUSY in our construction is in agreement with

geometrical considerations.

Such a correspondence was partly suggested in [71] and also discussed in [72] in relation

to the gauged WZW model of the GMM type [73].

Additionally we note that T 1,1 is isomorphic with the Stiefel manifold V2(R4) and have

the U(1)-fibration

T 1,1 U(1)−→ CP1 × CP1 , (3.25)

which gives the geometric interpretation of (3.22) for the case p = q = 1.

19

3.4 Coset Constructions of d = 3 Superstring Vacua

Since we are now assuming dimG/H = 6, the condition for the spacetime SUSY (2.15)

reduces to the simple criterion given in [46]: The spacetime SUSY exists if and only if it

holds that

M =Gk × SO(6)1

H∼= Gk × SU(3)1

H× SO(6)1

SU(3)1

∼= Gk × SU(3)1

H× U(1)3/2 , (3.26)

as a well-defined coset CFT. Here we used the equivalence

SO(6)1/SU(3)1∼= U(1)3/2 . (3.27)

We first focus on the examples when G is a compact simple group. The possible coset

spaces are listed as follows;

G/H = SO(7)/SO(6) , SU(4)/(SU(3)× U(1)) , G2/SU(3) ,

SO(5)/(SO(3)× U(1)) , SU(3)/(U(1)× U(1)) . (3.28)

We also pick up the following example of a semi-simple G

G/H = (SU(2)× SU(2)× SU(2))/SU(2) . (3.29)

1. G/H = SO(7)/SO(6) , SU(4)/(SU(3)× U(1))

The corresponding supercoset CFT’s are defined by

M =SO(7)k × SO(6)1

SO(6)k+1, M =

SU(4)k × SO(6)1

SU(3)k+1 × U(1)6(k+4)

. (3.30)

Clearly, the denominator groups are “too big” to allow the rearrangement (3.26). We

thus find no vacua with spacetime SUSY.

2. G/H = G2/SU(3)

In this case we find

M =(G2)k × SO(6)1

SU(3)k+1

∼= (G2)k × SU(3)1

SU(3)k+1× U(1)3/2 . (3.31)

We thus obtain a supersymmetric vacuum corresponding to a non-compact G2 holonomy

manifold. This model has been studied in [46].

20

This coset space is isomorphic with S6 and it is a typical example of a nearly Kahler

but non-Kahler manifold [14]. Our CFT result is in accordance with this fact.

3. G/H = SO(5)/(SO(3)× U(1))

We have two possibilities of the embedding of SO(3)× U(1); one is to embed SO(3)

along the long root of SO(5), and another is along its short root. U(1) has to be embedded

in the direction orthogonal to SO(3) in each case.

(i) SO(3) embedded along a long root :

Suppose SO(3) is embedded along the long root of SO(5). The coset space is

topologically isomorphic with S7/U(1) ∼= CP3 and can be endowed with a (non-

HSS) parabolic structure.6 Hence M is a (non-HSS) Kazama-Suzuki model. Based

on the formulas (3.3) we obtain

M =SO(5)k × SO(6)1

SO(3)k+12× U(1)k+3

∼= SO(5)k × SO(6)1

SU(2)k+1 × U(1)k+3

∼= SO(5)k × SU(3)1

SU(2)k+1 × U(1)k+3

× U(1)3/2 . (3.32)

We thus obtain superstring vacua of non-compact G2-holonomy manifolds.

The coset space defined here can be equipped with an Einstein metric that is SO(5)

invariant, but not compatible with a definite complex structure. This is an example

of a nearly Kahler manifold and gives rise to the well-known solution with the G2-

holonomy metric on its cone [74, 75] (see also [5]). Our result seems consistent with

this fact.

(ii) SO(3) embedded along a short root :

If SO(3) is embedded along the short root, the coset space is the Grassmannian

G2(R5), which is an HSS. We find from (3.3)

M =SO(5)k × SO(6)1

SO(3)k+2 × U(1)2(k+3)

. (3.33)

In this case we cannot rewrite it as in (3.26). The vacua have no SUSY.6The most familiar coset realization of CP3 is of course SU(4)/(SU(3)× U(1)), which is a hermitian

symmetric space (HSS). The non-HSS Kahler coset SO(5)/(SO(3)×U(1)) gives an inequivalent parabolic

decomposition

g = h + m+ + m− , [h,m±] ⊂ m± , [m±,m±] ⊂ m± ,

with m+, m− being non-abelian.

21

4. G/H = SU(3)/(U(1)× U(1))

We must specify the charge lattice Γ associated to the U(1)×U(1) action to define the

supercoset CFT M, although the corresponding coset space is always isomorphic with

the flag manifold F1,2(C3) irrespective of the choice of Γ. Let Q be the root lattice of

SU(3). According to (3.3), Γ must be a sublattice of√k + 3Q, where k is the level of

SU(3), in order to maintain the worldsheet SUSY. On the other hand, the criterion for

spacetime SUSY (3.26) requires that the coset

SU(3)k × SU(3)1

U(1)× U(1)(3.34)

is well-defined, which implies the condition Γ ⊂ √kQ ⊕ Q. Especially, we cannot adopt

the simplest choice Γ =√k + 3Q for a generic k.

The charge lattices Γ which admit both worldsheet and spacetime SUSY are con-

structed as follows: Let αi, βi (i = 1, 2) be the simple roots corresponding to the current al-

gebras SU(3)k, SU(3)1 respectively, normalized as α2i = β2

i = 2, (α1, α2) = (β1, β2) = −1,

(αi, βj) = 0. We set

ν1 =√kα1 + β1 + 2β2 ,

ν2 =√k(α1 + 2α2) + 3β1 , (3.35)

and define the lattice

Λ = Zν1 + Zν2 . (3.36)

They satisfy

(ν1, ν1) = 2(k + 3) , (ν2, ν2) = 6(k + 3) , (ν1, ν2) = 0 . (3.37)

By definition Λ is a sublattice of√kQ⊕Q, and by means of the identification

√k + 3 γ1 = ν1 ,

√k + 3 (γ1 + 2γ2) = ν2 , (3.38)

where γ1, γ2 denote the simple roots corresponding to the total current SU(3)k+3, we can

also regard Λ as a sublattice of√k + 3Q. More precise argument is given as follows:

Recall that the total Cartan current of SU(3)k+3 is decomposed as [56, 76]

(t, H(z)) = (t, H(z)) +∑

α∈∆+

〈α, t〉 : ψαψ−α : , (3.39)

where H is the bosonic current and ψα are the free fermions along the coset direction.

(ψα(z)ψα′(0) ∼ δα+α′,0/z). ∆+ denotes the set of positive roots. On the other hand, the

22

embedding SU(3)1 ⊂ SO(6)1 is given by defining the SU(3)1 current K(z) from the 6

free fermions χi, χ∗i (i = 1, 2, 3), (χ∗i (z)χj(0) ∼ δij/z, χi(z)χj(0) ∼ χ∗i (z)χ

∗j (0) ∼ 0);

(Eii − Ejj, K(z)) =: χ∗iχi : − : χ∗jχj : , (Eij , K(z)) = χ∗iχj (i 6= j) , (3.40)

where we set (Eij)ab = δiaδjb. Identifying the coset fermions ψα and χi, χ∗j by the relation

χ1 = ψ−θ , χ∗1 = ψθ ,

χ2 = ψα2 , χ∗2 = ψ−α2 ,

χ3 = ψα1 , χ∗3 = ψ−α1 , (3.41)

we find that

(λ3, H) = (λ3, H) + (: χ∗1χ1 : + : χ∗2χ2 : −2 : χ∗3χ3 :) ,

(λ8, H) = (λ8, H) +√

3(: χ∗1χ1 : − : χ∗2χ2 :) , (3.42)

where λ3, λ8 are the Gell-Mann matrices corresponding to α1,1√3(α1 + 2α2);

λ3 =

1

−1

0

, λ8 =1√3

1

1

−2

. (3.43)

These relations justify the identification (3.35), (3.38).

We should also take account of the symmetry by the Weyl group W . Therefore, the

charge lattice Γ gives SUSY, if (and only if) it is a sublattice of w · Λ with some Weyl

reflection w ∈W . Especially, in the simplest case Γ = Λ, we obtain

M =SU(3)k × SO(6)1

U(1)k+3 × U(1)3(k+3)

∼= SU(3)k × SU(3)1

U(1)k+3 × U(1)3(k+3)

× U(1)3/2 . (3.44)

One can directly check that the coset part in the last line is actually well-defined because

of the relation SU(3)1 ∼ U(1)1×U(1)3 ∼ U(1)9×U(1)3. In conclusion, we have obtained

an infinite family of supersymmetric vacua of G2-holonomy corresponding to the charge

lattice Γ mentioned above.

We have a comment: The flag manifold F1,2(C3) has the standard Kahler-Einstein

metric, which is not SU(3)-invariant. However, this space is known to have a second

(non-Kahler) Einstein metric that is SU(3) invariant and compatible with a nearly Kahler

23

structure [14] (see also [5]). There exists a well-known solution of G2-holonomy metric

on this cone [74, 75]. Our CFT result again seems to be consistent with the classical

geometry. However, while the different choice of Γ leads to the same homogeneous space

F1,2(C3) classically, we obtain inequivalent string vacua depending on the choice of Γ in

our coset CFT construction.

We further consider examples of a non-simple G, motivated by the example of a nearly

Kahler space S3 × S3.

5. G/H = (SU(2)× SU(2)× SU(2))/SU(2)

We have three ways of SU(2) embedding. In all the three cases the coset space is

topologically isomorphic with S3 × S3.

(i) SU(2) embedded only in an SU(2)-factor :

This case is very easy, since one of the SU(2)-factors in the numerator is canceled.

The relevant supercoset reduces to

M = SU(2)k1 × SU(2)k2 × SO(6)1 . (3.45)

This obviously gives rise to supersymmetric vacua with 16 supercharges.

If we further make an S1 compactification, this model will be converted by (2.33)

into the background AdS3× S3× S3× S1 discussed in [77], which realizes the large

N = 4 superconformal symmetry on the boundary of AdS3.

(ii) SU(2) embedded in two SU(2)-factors :

In this case the relevant supercoset becomes

M =SU(2)k1 × SU(2)k2 × SO(3)1

SU(2)k1+k2+2

× SU(2)k3 × SO(3)1 . (3.46)

It leads to non-supersymmetric string vacua.

(iii) SU(2) embedded in all of the SU(2)-factors :

This third case is the most interesting. The relevant supercoset is defined as

M =SU(2)k1 × SU(2)k2 × SU(2)k3 × SO(6)1

SU(2)k1+k2+k3+4

. (3.47)

24

In the same way as before the existence of spacetime SUSY is confirmed by the

equivalence

M ∼= SU(2)k1 × SU(2)k2 × SU(2)k3 × SU(3)1

SU(2)k1+k2+k3+4× U(1)3/2 . (3.48)

Recall the conformal embedding SU(2)4 ⊂ SU(3)1 is associated with the D4-type

modular invariant. Namely, we have the character relations

χSU(3)1basic (τ) = χ

(4)0 (τ) + χ

(4)4 (τ)

χSU(3)1fund (τ) = χ

SU(3)1

fund(τ) = χ

(4)2 (τ) , (3.49)

where χ(k)` denotes the SU(2)k character of spin `/2. Accordingly, the coset part in

(3.48) is really well-defined.

This type string vacua of G2-holonomy are regarded as natural generalizations of

those given in [45]. In fact, the special cases of k1 = k, k2 = k3 = 0 reduces to

Rφ × ψφ × SU(2)k × SO(6)1

SU(2)k+4

∼= Rφ × ψφ × SU(2)k × SU(2)2

SU(2)k+2× SU(2)k+2 × SU(2)2

SU(2)k+4

∼= Rφ × ψφ ×MN=1k+2 ×MN=1

k+4 , (3.50)

where MN=1m denotes the m-th N = 1 minimal model (c = 3

2− 12

m(m+2)). These are

precisely the models constructed in [45].

We further make a comment: As mentioned in [5], the trivial round sphere metric

on S3 × S3 does not generate the G2-holonomy on its cone. While, the second

Einstein metric based on the identification S3 × S3 ∼= (SU(2))3/SU(2) leads to a

G2-holonomy on its cone and there exists a well-known G2 metric [74, 75]. It seems

plausible to relate the former case with the SCFT (3.45) and the latter with (3.47),

although the precise interpretation of this relation is yet unclear.

To complete our classification of the d = 3 string vacua, let us also consider the

diagonal coset (3.12). Although dimG/H 6= 6, we obtain the supersymmetric vacua of

G2-holonomy, by restricting the levels of current algebras. In fact, if we set k2 = 2 (or

k1 = 2), the model again reduces to the one just considered (3.50). It is easy to see that

any other choices of levels do not lead to supersymmetric vacua.

25

3.5 Coset Constructions of d = 2 Superstring Vacua

The d = 2 cases are the most involved because we have three possibilities of supersym-

metric string vacua (except for the case of trivial flat space), that is, the Sp(2), SU(4)

and Spin(7)-holonomies, each of which corresponds to a tri-Sasakian, Sasaki-Einstein,

and weak G2 base spaces, respectively.

The 7-dimensional Einstein homogeneous spaces are completely classified in [22]. We

first focus on the cases of simple G listed as

G/H = SO(8)/SO(7) , SO(7)/G2 , SU(4)/SU(3) ,

SU(3)/U(1) , SO(5)/SO(3) . (3.51)

We further discuss a few cases with a non-simpleG; G/H = (SU(3)×SU(2))/(SU(2)×U(1)), G/H = (SU(2)×SU(2)×SU(2))/(U(1)×U(1)), which are also found in the list of

[22] (see also [24]). The other example is again (3.12). Although dimG/H 6= 7 here, it will

turn out that the Spin(7)-holonomy vacua are also obtained under a suitable restriction

of levels, as in the d = 3 case.

The SUSY condition (2.17) now reduces to a criterion [46];

M =Gk × SO(7)1

H

∼= Gk × (G2)1

H× tri-critical Ising , (3.52)

by using the identification

SO(7)1/(G2)1∼= tri-critial Ising , (3.53)

which was first pointed out in [47].

1. G/H = SO(8)/SO(7) :

We have no spacetime SUSY in this case, because the denominator group is too large

to make the rearrangement (3.52) possible.

2. G/H = SO(7)/G2 :

This model gives the superstring vacua of Spin(7)-holonomy studied in [46]. The

condition (3.52) is easily checked as follows;

M =SO(7)k × SO(7)1

(G2)k+1

∼= SO(7)k × (G2)1

(G2)k+1× SO(7)1

(G2)1

∼= SO(7)k × (G2)1

(G2)k+1× tri-critical Ising . (3.54)

26

The trivial case k = 0 (i.e. M = tri-critical Ising) corresponds to the model discussed in

[45].

3. G/H = SU(4)/SU(3) :

We can similarly prove that the condition (3.52) is satisfied here. However, based on

(2.7), we can further show that the worldsheet SUSY is enhanced to N = 2;

Rφ × ψφ ×M = Rφ × ψφ × SU(4)k × SO(7)1

SU(3)k+1

∼= [Rφ × ψφ × U(1)6(k+4) × SO(1)1

]× SU(4)k × SO(6)1

SU(3)k+1 × U(1)6(k+4)

.(3.55)

The part [· · ·] is interpreted as the N = 2 Liouville theory. The criticality condition (2.3)

gives Q2φ = 12/(k + 4), and hence U(1)6(k+4) describes precisely the compact boson of

N = 2 Liouville theory. The remaining coset CFT is the Kazama-Suzuki model for CP3.

Therefore, these string vacua correspond to non-compact CY4 manifolds.

4. G/H = SO(5)/SO(3) :

This example is quite amazing. We find that all of the three possible holonomies

Sp(2), SU(4), and Spin(7) are realized.

(i) SO(3) embedded along a long root :

Suppose SO(3) is embedded along a long root of SO(5), in other words, embedded

in one of the SO(3)’s of the SO(3) × SO(3)(∼= SO(4)) subgroup of SO(5). This

coset space is isomorphic with S7, and as is obvious by construction, we have a

remaining SO(3)(' SU(2)) symmetry. This space is an elementary example of the

tri-Sasakian manifold and can be regarded as an SU(2)-bundle over a Wolf space:

M = SO(5)/SO(3) ∼= S7 SU(2)−→ M/SU(2) ∼= S4 . (3.56)

This is nothing but the familiar (quaternionic) Hopf fibration.

In terms of the coset CFT, we obtain

Rφ × ψφ ×M = Rφ × ψφ × SO(5)k × SO(7)1

SO(3)k+12

∼= Rφ × ψφ × SO(5)k × SO(7)1

SU(2)k+1

∼= [Rφ × ψφ × SU(2)k+1 × SO(3)1

]× SO(5)k × SO(4)1

SU(2)k+1 × SU(2)k+1.

(3.57)

27

The criticality condition (2.3) yields Q2φ = 8/(k + 3). Since the supercoset part of

the last line is associated to a Wolf space, the worldsheet SUSY should be enhanced

to N = 4, as we already discussed. In this way, we have obtained the superstring

vacua corresponding to the Sp(2)-holonomy. The SUSY cancellation reduces to the

identity (2.10).

(ii) SO(3) embedded along a short root :

Suppose SO(3) is embedded along a short root, in other words, embedded diagonally

into the SO(3)×SO(3) subgroup. This is found to be the “canonical embedding” so

that the vector representation of SO(5) is decomposed as 5 → 3+1+1. Hence the

coset space is isomorphic with the Stiefel manifold V2(R5), which has the canonical

U(1)(∼= SO(2))-fibration over the Grassmannian G2(R5) (an example of HSS);

M = SO(5)/SO(3) ∼= V2(R5)

U(1)−→ M/U(1) ∼= G2(R5) . (3.58)

This is a typical example of the Sasaki-Einstein homogeneous space. In terms of

the coset CFT we thus find that

Rφ × ψφ ×M = Rφ × ψφ × SO(5)k × SO(7)1

SO(3)k+2

∼= [Rφ × ψφ × U(1)2(k+3) × SO(1)1

]× SO(5)k × SO(6)1

SO(3)k+2 × U(1)2(k+3)

.(3.59)

We here obtain Q2φ = 9/(k + 3), resulting in 32 · 2/Q2

φ = 2(k + 3). Therefore,

U(1)2(k+3)-factor exactly reproduces the N = 2 Liouville theory. The remaining

coset CFT is the Kazama-Suzuki model associated to G2(R5). In this way we find

that the total system has N = 2 worldsheet SUSY and SU(4)-holonomy.

(iii) SO(3) embedded as a maximal subgroup of SO(5):

The third case is the most non-trivial. Consider the embedding of SO(3) along

2α1 + 3α2, where α1, α2 are the long and short roots of SO(5) (α21 = 2, α2

2 = 1,

α1 · α2 = −1). The simple root of SO(3) is defined as the projection of the highest

root of SO(5) and thus identified as θ′ ≡ 15(2α1 + 3α2). In this embedding the

adjoint representation of SO(5) is decomposed as 10 → 3 + 7, which means it

is a maximal embedding (see [22] for the detail). Since we have (θ′, θ′) = 1/5, the

relevant supercoset should be

M =SO(5)k × SO(7)1

SO(3)5k+14. (3.60)

28

Since we have no remaining symmetry of SU(2) or U(1) in this coset, it is obvious

that the worldsheet SUSY cannot be enhanced. We thus obtain at most 2 super-

charges in spacetime corresponding to the Spin(7)-holonomy. The criterion for the

spacetime SUSY (3.52) is now expressed as

M ∼= SO(5)k × (G2)1

SO(3)5k+14× SO(7)1

(G2)1

∼= SO(5)k × (G2)1

SU(2)10k+28

× tri-critical Ising , (3.61)

and we require that the coset CFT in the last line should be well-defined. Especially,

we must ask whether we can consistently define (G2)1/SU(2)28. We first note that

(G2)1 and SU(2)28 have the equal central charge c = 14/5. So, (G2)1/SU(2)28

would be a topological coset CFT, if it is well-defined. It is actually known that the

conformal embedding SU(2)28 ⊂ (G2)1 exists (see for example, [69])7. More precise

relation is as follows: SU(2)28 is known to have the E8-type modular invariant [70],

in which the partition function is given as

Z =∣∣∣χ(28)

0 + χ(28)10 + χ

(28)18 + χ

(28)28

∣∣∣2 +∣∣∣χ(28)

6 + χ(28)12 + χ

(28)16 + χ

(28)22

∣∣∣2 .(3.62)

This partition function is in fact equivalent to the diagonal modular invariant of

(G2)1 due to the character relations;

χ(G2)1basic(τ) =

(χ

(28)0 + χ

(28)10 + χ

(28)18 + χ

(28)28

)(τ) ,

χ(G2)1fund (τ) =

(χ

(28)6 + χ

(28)12 + χ

(28)16 + χ

(28)22

)(τ) . (3.63)

Accordingly, the rewriting (3.61) is actually well-defined. We have achieved the

string vacua of Spin(7)-holonomy.

5. G/H = SU(3)/U(1) :

To define the supercoset CFT, we have to specify the U(1)-embedding, which is char-

acterized by the 1-dimensional charge lattice Γ ⊂ √k + 3Q, where Q = Zα1 + Zα2 is the

root lattice of SU(3). ((αi, αi) = 2, (α1, α2) = −1.)

7The explicit embedding of SU(2) ⊂ G2 here is given as follows: The simple root of SU(2) is identified

with θ′ = 114 (α1 + 6α2), where α1, α2 denote the long and short roots of G2 (α2

1 = 2, α22 = 2/3,

α1 · α2 = −1). By this embedding, the adjoint representation of G2 is decomposed as 14 → 3 + 11.

Hence, this is also the maximal embedding. The square length of θ′ is equal 1/14, which is compatible

with the existence of coset CFT (G2)1/SU(2)28.

29

The spacetime SUSY requires that the following rewriting should be possible; 8

M =SU(3)k × SO(7)1

U(1)

∼= SU(3)k × SU(3)1

U(1)× SO(1)1 × U(1)3/2 . (3.64)

As we discussed in the d = 3 analysis, this rewriting is possible if and only if Γ is generated

by an element µ1 of the lattice w ·Λ, where Λ is defined in (3.36) and w is a Weyl reflection.

However, we can show that√k + 3Q = W · Λ. So, we have the spacetime SUSY for an

arbitrary choice of µ1.

On the other hand, the N = 4 enhancement of worldsheet SUSY occurs in the special

cases µ1 = m(w · ν2)(≡ m(w · (α1 + 2α2))), where m is an arbitrary integer and w is a

Weyl reflection. In fact, (only) in that case we can find an SU(2) symmetry along the

transverse direction µ2 = w · ν1(≡ w · α1), resulting in the equivalence

Rφ × ψφ ×M ∼= [Rφ × ψφ × SU(2)k+1 × SO(3)1

]× SU(3)k × SO(4)1

U(1)3(k+3) × SU(2)k+1.(3.65)

The coset part is associated to CP2, which possesses the structures of both the HSS and

Wolf space. Hence the worldsheet SUSY is enhanced to N = 4.

As a consistency check, we can also check the N = 2 structure on the worldsheet. The

following rewriting is also possible;

Rφ × ψφ ×M ∼= [Rφ × ψφ × U(1)k+3 × SO(1)1

]× SU(3)k × SO(6)1

U(1)3(k+3) × U(1)k+3. (3.66)

The part [· · ·] describes the N = 2 Liouville since Q2φ = 8/(k + 3), and the coset part is

the Kazama-Suzuki model considered in the d = 3 analysis.

It is also obvious that, if we cannot write µ1 as the form µ1 = m(w ·ν2), the worldsheet

SUSY cannot be enhanced for generic values of k.

In summary, we have shown that

(i) If Γ has the form Γ = Z(mw · ν2), with some Weyl reflection w and an integer m,

we have the N = 4 worldsheet SUSY, and the string vacua corresponds to an

Sp(2)-holonomy.

(ii) If Γ does not, we have the supersymmetric vacua with 4 supercharges, but the world-

sheet SUSY is at most N = 1. In this case, the coset CFT M has a remaining

U(1)-symmetry and seems to correspond to a compactification on a space of the

8This is a slightly stronger condition than (3.52) and leads to twice as many supercharges as compared

to the Spin(7)-holonomy case (i.e. the same number of supercharges as the SU(4) and G2-holonomy).

However, it is easy to show that (3.52) inevitably reduces to (3.64) in this case.

30

form S1 × G2-manifold. However, the U(1)-charge is not independent of the quan-

tum numbers in the remaining sector and this is some kind of an orbifold space.

Let us make a few comments: The coset space SU(3)/U(1) is known under the name

Aloff-Wallach space [25] and written as N(m, `) 9 with the U(1)-action

eiθ 7−→

eimθ 0 0

0 ei`θ 0

0 0 e−i(m+`)θ

(3.67)

where m, ` are relatively prime integers. They are not diffeomorphic for different param-

eters m, ` (unless we have a Weyl reflection connecting them).

It is known that the spaces N(m, `) can be endowed with two types of Einstein metrics

[19, 20], denoted as N(m, `)I and N(m, `)II (the “squashed” one) in [24]. The generic cases

of (m, `) 6= (1, 1) become weak G2 manifolds irrespective of the choice of Einstein metrics.

On the other hand, N(1, 1)I is known to be tri-Sasakian (and also Sasaki-Einstein) [19, 12],

while N(1, 1)II has the weak G2 holonomy [20, 12]. The choice of m, ` is in one-to-one

correspondence with the charge lattice Γ introduced above. The first case (i) corresponds

to N(1, 1) and leads to theN = 4 worldsheet SUSY, while the second case (ii) corresponds

to the cases (m, `) 6= (1, 1) and at most the N = 1 worldsheet SUSY is allowed. In this

sense our algebraic construction agrees with the geometrical analysis. The amount of

worldsheet SUSY is exactly as expected. Among other things, N(1, 1)I has the SU(2)-

fibration characteristic of the tri-Sasakian homogeneous space;

N(1, 1)ISU(2)−→ CP2 , (3.68)

which corresponds to (3.65). It also has the U(1)-fibration for the Sasaki-Einstein space;

N(1, 1)IU(1)−→ F1,2(C

3) , (3.69)

where we should regard the flag manifold F1,2(C3) as a Kahler-Einstein space. This of

course corresponds to (3.66).

We will later discuss the CFT interpretation of the squashed geometry of N(1, 1)II.

9In many literature it is also denoted as Npqr [19], where Npqr =SU(3)× U(1)U(1)× U(1)

with the integer

parameters p, q, r characterizing the remaining U(1) symmetry as

Z = pi√

32

λ8 + qi

2λ3 + riY .

(λ3, λ8 are the Gell-Mann matrices (3.43) and Y is the generator of U(1) in the numerator.) In particular,

N(1, 1) is equal to N010.

31

We further analyse a few examples with a non-simple group G.

6. G/H = (SU(3)× SU(2))/(SU(2)× U(1))

We have various possiblities of embedding of SU(2)× U(1) as listed in [22].

(i) SU(2) embedded as the isospin subgroup :

The relevant coset SCFT should have the form

M =SU(3)k1 × SU(2)k2 × SO(7)1

SU(2)k1+1 × U(1). (3.70)

To define the model completely, we still have to fix the U(1) embedding. Choosing

the isospin SU(2) subgroup along the simple root α1 (simple roots of SU(3) : α1,

α2 with α21 = α2

2 = 2, α1 · α2 = −1, as usual), let us introduce the following charge

lattice for U(1)-action;

Γ = Z

(q√k1 + 3(α1 + 2α2)− p′

√k2 + 2β

), (3.71)

where β is the simple root of the SU(2) factor (β2 = 2). The U(1) action generated

by Γ obviously commutes with SU(2)isospin for arbitrary q, p′. The theta function

associated to this charge lattice yields the U(1) factor U(1)3(k1+3)q2+(k2+2)p′2 in (3.70),

and we obtain

M ∼= SU(3)k1 × SO(4)1

SU(2)k1+1 × U(1)3(k1+3)

× SU(2)k2 × SO(2)1

U(1)k2+2

× SO(1)1 × U(1)(k1+3)(k2+2){3q2(k1+3)+p′2(k2+2)} , (3.72)

using the relation

U(1)3(k1+3) × U(1)k2+2

U(1)3(k1+3)q2+(k2+2)p′2∼= U(1)3(k1+3)(k2+2){3q2(k1+3)+p′2(k2+2)} , (3.73)

of the product formula of theta functions. The first coset part in (3.72) is the

Kazama-Suzuki model for CP2 and the second coset is the N = 2 minimal model

of level k2. On the other hand, the criticality condition (2.3) leads to

Q2φ =

2(k1 + 3k2 + 9)

(k1 + 3)(k2 + 2). (3.74)

We thus need the factor U(1)(k1+3)(k2+2)(k1+3k2+9) in (3.72) to get theN = 2 Liouville

sector. Therefore, the worldsheet SUSY is enhanced to N = 2, if and only if p′ = 3q

holds, which yields a string vacuum of SU(4)-holonomy.

32

In the case of p′ 6= 3q, we have at mostN = 1 worldsheet SUSY. Then, the spacetime

SUSY requires that the following rewriting should be possible (as in the analysis of

SU(3)/U(1));

M ∼= SU(3)k1 × SU(2)k2 × SU(3)1

SU(2)k1+1 × U(1)3q2(k1+3)+p′2(k2+9)

× SO(1)1 × U(1)3/2 . (3.75)

For generic values of k1, k2, this is possible only for p′ = 3q, which goes back to the

N = 2 case already discussed. In this way, we conclude that this type string vacua

are supersymmetric if and only if p′ = 3q holds, and in those cases we obtain the

SU(4)-holonomy.

We have a comment in connection to known results in Kaluza-Klein SUGRA: This

case corresponds to the homogeneous space with SU(3) × SU(2) × U(1) isometry

first studied in [17];

Mpqr =SU(3)× SU(2)× U(1)

SU(2)× U(1)× U(1), (3.76)

where SU(2) is embedded as the isospin subgroup of SU(3) and the integer param-

eters p, q, r characterize the remaining U(1)-symmetry as

Z = pi√

3

2λ8 + q

i

2σ3 + riY , (3.77)

where λ8 is the Gell-Mann matrix (the generator transverse to the SU(2)isospin)

(3.43), σ3 is the Pauli matrix and Y is the generator of the U(1)-factor in the

numerator. Our coset CFT M with the charge lattice Γ is naturally associated to

the space Mpq0 under the identification p′ = 3p (at least for the cases k1+3 = k2+2).

It is known [21] that every coset space Mpqr can be equipped with Einstein metrics

and allows the spacetime SUSY only in the case p = q. M110 is a regular Sasaki-

Einstein space and Mppr is regarded as an orbifold of it. The U(1)-fibration for the

Sasaki-Einstein space M110 is written as

M110 U(1)−→ CP2 × CP1 , (3.78)

which corresponds to (3.72) for q = 1, p′ = 3. These aspects fit nicely with our

algebraic construction.

(ii) SU(2) embedded in the explicit SU(2) factor :

Obviously, this reduces to the case of SU(3)/U(1) we studied previously.

33

(iii) SU(2) embedded in both of SU(3) and the explicit SU(2) factor :

In this case the only possibility is the diagonal embedding SU(2) ⊂ SU(2)isospin ×SU(2) ⊂ SU(3)× SU(2). The U(1)-factor must necessarily be embedded along the

λ8 ∝ α1 + 2α2 direction. The corresponding coset SCFT is defined as

M =SU(3)k1 × SU(2)k2 × SO(7)1

SU(2)k1+k2+3 × U(1)3(k1+3)

. (3.79)

We can rewrite it as

M ∼= SU(3)k1 × SU(2)k2 × (G2)1

SU(2)k1+k2+3 × U(1)3(k1+3)× tri-critial Ising . (3.80)

(Recall that (G2)1 ∼ SU(2)3 × U(1)1 ∼ SU(2)3 × U(1)9.) Therefore, we have

obtained a string vacuum with Spin(7)-holonomy.

The coset space considered here is topologically isomorphic with N(1, 1), and this

string vacuum is supposed to correspond to the squashed geometry of N(1, 1) men-

tioned before. Under the limit k2 → ∞, the SU(2)-factors in (3.79) are canceled

out, and we recover the N(1, 1) coset SCFT M =SU(3)k × SO(7)1

U(1)3(k+3)

.

(iv) SU(2) embedded as a maximal subgroup of SU(3) :

In this case U(1) has to be embedded only in the SU(2) factor. As in (3.17), the

relevant supercoset CFT is written as

M =SU(3)k1 × SU(2)k2 × SO(7)1

SU(2)4k1+10 × U(1)k2+2(3.81)

∼= SU(3)k1 × SO(5)1

SU(2)4k1+10× SU(2)k2 × SO(2)1

U(1)k2+2.

The second line corresponds to the fact that this coset space is topologically iso-

morphic with S5 × S2 (see [22]), and no remaining U(1) symmetry exists. As is

easily shown, we have no spacetime SUSY in this case. It is again consistent with

the known results of Kaluza-Klein SUGRA [22].

7. G/H = (SU(2)× SU(2)× SU(2))/(U(1)× U(1)) :

This model may regarded as a natural generalization of the d = 4 vacuum (SU(2) ×SU(2))/U(1) and also the CHS σ-model. The supercoset has the form

M =SU(2)k1 × SU(2)k2 × SU(2)k3 × SO(7)1

U(1)× U(1). (3.82)

34

Similarly to the d = 4 case, we try to rewrite as

Rφ × ψφ ×M ∼ [Rφ × ψφ × U(1)× SO(1)1

]×Mk1 ×Mk2 ×Mk3 , (3.83)

where Mk ≡ SU(2)k × SO(2)1

U(1)k+2denotes the N = 2 minimal model again. Since we obtain

Q2φ =

2(N2N3 +N3N1 +N1N2)

N1N2N3, (Ni ≡ ki + 2) , (3.84)

from the criticality condition (2.3), the criterion for the part [· · ·] to become the N = 2

Liouville is whether we can factorize U(1)N1N2N3(N2N3+N3N1+N1N2). Namely, we want to

derive a relation

U(1)N1 × U(1)N2 × U(1)N3

U(1)× U(1)∼= U(1)N1N2N3(N2N3+N3N1+N1N2) , (3.85)

by a suitable choice of the charge lattice Γ of U(1)×U(1). To describe it explicitly, let αi

(i = 1, 2, 3) be the simple roots of each SU(2) factors, normalized as (αi, αj) = 2δij. The

two dimensional lattice Γ must be defined as a sublattice of Z√N1α1+Z

√N2α2+Z

√N2α2.

Introducing integer parameters p, q, r, we set

ν1 = q√N1α1 − p

√N2α2 ,

ν2 = prN2N3

√N1α1 + qrN3N1

√N2α2 −N3(p

2N2 + q2N1)√N3α3 ,

ν3 = pN2N3

√N1α1 + qN3N1

√N2α2 + rN1N2

√N3α3 . (3.86)

Then we find that they are orthogonal to each other, and also,

(ν3, ν3) = 2N1N2N3(p2N2N3 + q2N3N1 + r2N1N2) . (3.87)

If we choose Γ as (a sublattice of) Zν1 + Zν2, we obtain the theta function identity such

as

Θ∗,N1(τ)Θ∗,N2(τ)Θ∗,N3(τ) =∑

Θ(Γ)∗ (τ)Θ∗,N1N2N3(p2N2N3+q2N3N1+r2N1N2) . (3.88)

Accordingly, if and only if p = q = r holds, the wanted relation (3.85) is obtained and

hence the worldsheet SUSY enhances to N = 2. We find the string vacuum with SU(4)-

holonomy in this case. It is also not difficult to see that the spacetime SUSY cannot

exist in other cases as in the previous analysis. The special cases of k1 = k, k2 = k3 = 0

correspond to the CY4 with the Ak+1-type singularity studied in [29, 34, 71].

We also make a comment in connection to the Kaluza-Klein SUGRA: Consider the

Einstein homogeneous space [18]

Q(p, q, r) =SU(2)× SU(2)× SU(2)

U(1)× U(1), (3.89)

35

where p, q, r parameterize the remaining U(1)-symmetry as before. It is known that we

have the spacetime SUSY only in the cases p = q = r and the Q(p, q, r) space becomes a

Sasaki-Einstein space in this case. For the cases of N1 = N2 = N3, the above choice of

charge lattice Γ precisely reproduces the coset space Q(p, q, r) (the vector ν3 describes the

remaining U(1) symmetry), and thus the SUSY condition coincides precisely. Moreover,

the U(1)-fibration

Q(p, q, r)U(1)−→ CP1 × CP1 ×CP1 (3.90)

is naturally related with (3.83). Our CFT analysis again is consistent with that of Kaluza-

Klein SUGRA.

8. G/H = (SU(2)× SU(2))/SU(2) :

Finally we present an example with dimG/H 6= 7. Consider again the N = 1 diagonal

coset (3.12) and set k1 = k, k2 = 1;

M =SU(2)k × SU(2)1 × SO(3)1

SU(2)k+3

. (3.91)

Since dimG/H = 3 rather than dimG/H = 7, we use a different relation to present the

spacetime SUSY;

SU(2)1 × SU(2)2

SU(2)3

∼= tri-critical Ising . (3.92)

We then obtain

M ∼= SU(2)k × SU(2)3

SU(2)k+3× SU(2)1 × SU(2)2

SU(2)3

∼= SU(2)k × SU(2)3

SU(2)k+3× tri-critial Ising , (3.93)

which satisfies the SUSY condition (2.17). In this way we obtain superstring vacua of

Spin(7)-holonomy manifolds. Simplest case k = 0 again reduces to the model given in

[45]. Since dimG/H 6= 7, it seems difficult to relate these string vacua with the solutions

of SUGRA.

Before closing this section we want to make a few comments on the relation between

our CFT and the classical geometry of special holonomy manifolds based on the cone

construction.

1. The comparison between the geometrical cones and our “CFT cones” leads to an

obvious disagreement when G/H is isomorphic with a round sphere S9−d. For the

36

d = 2 string vacua, for example, all the cosets G/H = SO(8)/SO(7), SO(7)/G2,

SU(4)/SU(3), SO(5)/SO(3) (the case when SO(3) is embedded along a long root)

are found to be topologically and metrically isomorphic with S7. The cone over S7

is of course the flat space R8, and hence allows the maximal spacetime SUSY. On

the other hand, the supercoset CFT’s based on them are really inequivalent with

each other, and yield string vacua with less spacetime SUSY. As we discussed above,

SO(8)/SO(7) leads to non-SUSY vacua, and SO(7)/G2, SU(4)/SU(3), SO(5)/SO(3)

provide Spin(7), SU(4), Sp(2) holonomies, respectively.

2. It is known that every 7-dim. tri-Sasakian manifold allows the second “squashed”

Einstein metric that provides a weak G2 holonomy [15, 13] (see also [12, 8]). A way

to present the squashing procedure is to replace the original tri-Sasakian coset G/H

byG× SU(2)

H × SU(2), which is topologically isomorphic with G/H.

For the case of N(1, 1) = SU(3)/U(1), the “squashed supercoset CFT” is de-

fined in (3.79). The similar construction is also possible for the tri-Sasakian coset

SO(5)/SO(3) (3.57), and could be identified as the “squashed S7” (often denoted

as J7). Namely, we deform (3.57) as

M =SO(5)k1 × SU(2)k2 × SO(7)1

SO(3)k1+12× SU(2)k1+k2+3

, (3.94)

where the SU(2) in the denominator is embedded diagonally into SU(2) × SU(2),

in which the first SU(2)-factor is the remaining one of SO(5)/SO(3) and the second

is the explicit SU(2)-factor. We can easily show that it leads to string vacua with

Spin(7)-holonomy in the similar manner as in the N(1, 1) case, and the original

tri-Sasakian coset (3.57) is recovered under the limit k2 → ∞.

The analogous relation is also found in the d = 3 example S3 × S3. The coset CFT

(3.47) may be regarded as the squashed version of (3.45). We again recover the

unsquashed one (3.45) in the limit k3 → ∞.

37

4 Marginal Deformations: Spectrum of Cosmological

Constant Operators

4.1 Cosmological Constant Operators Preserving Special Holon-

omy

Since the linear dilation CFT is singular as a worldsheet theory, we should introduce the

“cosmological constant operators” (Liouville potential terms) in order to eliminate its

singular behavior at the tip of the cone (Liouville exponential prevents the field φ going

out to −∞). In this section we consider cosmological constant terms which preserve

the spacetime supersymmetry and thus act as marginal perturbations in various models

discussed in previous sections, focusing in particular on the G2 and Spin(7) holonomy

cases.

As in the old days of two-dimensional gravity [78], the cosmological constant operator

is defined as the most relevant primary field of the “matter sector” multiplied by the

Liouville exponential. Here one might worry about the “c = 1 (c = 3/2) barrier”, since

the conformal matter Rd−1,1 ×M has the central charge bigger than 3/2 for any unitary

M. In fact if one considers an identity operator (of matter sector) multiplied by a Liou-

ville exponential, one finds the trouble of a complex Liouville exponent. This difficulty

is avoided in our case by the requirement of GSO projection for vacua with unbroken

spacetime SUSY. We should define the cosmological constant term for the most relevant

primary operator allowed by the GSO condition, and we can show that the Liouville expo-

nential is then always real as we shall see below. On the other hand in the case of broken

spacetime SUSY, it seems difficult to define a suitable cosmological operator that resolves

the singularity, since the most relevant operator becomes tachyonic and has a complex

Liouville exponential.

The general form of the marginal perturbation operator is written as follows;[G− 1

2,[G− 1

2, eγφO(NS)

M]], (4.1)

where O(NS)M denotes an NS primary field in the M sector. Here one can not choose

the identity operator O(NS)M = id. since the operator (4.1) then becomes mutually non-

local with respect to the spacetime SUSY operator (violates GSO condition). The BRST

invariance requires the on-shell condition

h(O(NS)M ) + h(eγφ) ≡ h(O(NS)

M )− 1

2γ2 − 1

2Qφγ = 12, (4.2)

under which (4.1) manifestly preserves the worldsheet N = 1 supersymmetry. To analyse

the spectrum of operators O(NS)M it is convenient to consider their supersymmetric partners

38

O(R)M in the Ramond sector, which have the conformal weight

h(O(R)M ) = h(O(NS)

M ) + 1− d16. (4.3)

This relation follows from the spacetime supersymmetry: Ramond states in the partition

functions are dressed by the spin fields of the Minkowski space and the Liouville fermion

and possess the same conformal weights as the NS states. Dimensions of the spin fields

add up to (d− 1)16 which accounts for the factor in (4.3).

The above relation may also be derived from the structure of our coset theories: NS

and R states in the coset theory have the general form,

O(NS)M = ΦΛ,s=2,λ [(G× SO(9− d))/H] ,

O(R)M = ΦΛ,s=1(−1),λ [(G× SO(9− d))/H] , (4.4)

where ΦΛ,s,λ [(G× SO(9− d))/H ] denotes a primary state in the coset (G × SO(9 −d))/H defined by the highest weights Λ, s and λ of the affine Lie algebras G, SO(9 −d) and H , respectively. s = 0, 2, 1,−1 stand for the basic,vector, spinor and cospinor

representations of SO(9 − d). Note that the weights Λ, λ of NS and R states are the

same for supersymmetric partners and thus the difference in their dimensions come from

that of the representations of SO(9 − d). Difference of spinor and vector dimensions

(9 − d)/16 − 1/2 = (1 − d)/16 accounts again for the RHS of (4.3). (In case a basic

representation s = 0 of the current algebra SO(9− d) is used in the NS state, there is an

additional factor +1/2 in the RHS of (4.3).)

The unitarity of M sector requires the inequality h(O(R)M ) ≥ cM/24, and thus we can

easily show that the Liouville exponent γ is always real with the help of the condition

(2.3).

Let us now fix the value of the exponent γ in the marginal perturbation operators (4.1).

From the old days of two-dimensional gravity it is known that Liouville exponentials have

different characteristics depending on whether γ > −Qφ/2 or γ < −Qφ/2 [54, 29, 31] (see

also [39, 40]).

• γ > −Qφ/2 : The operator eγφ is called non-normalizable, since the corresponding

wave function exponentially diverges at the asymptotic region φ → +∞. It is

interpreted as a coupling constant of the dual field theory, since the fluctuation

has a divergent kinetic energy. In the context of two dimensional gravity it is also

identified as the local scaling operators.

• γ < −Qφ/2 : The operator eγφ is called normalizable. The wave function is peaked

around the singular region φ→ −∞ as opposed to the above case. It is interpreted

as a VEV of the dynamical fields of the dual theory (the modulus of vacuum), since

the fluctuation has a finite kinetic energy.

39

Now we propose to choose the critical value for γ

γ = −Qφ/2 (4.5)

for our marginal operators (4.1). It corresponds to the maximal value of conformal weight

h(e−Qφ2

φ) = Q2φ/8 of the Liouville exponential with real γ and corresponds also to the

minimum value in the continuous spectrum of delta-function normalizable states γ =

−Qφ/2 + ip (p ∈ R). This situation is quite reminiscent of that of the c = 1 conformal

matter coupled to two dimensional gravity, or equivalently, the critical string on the

background of two dimensional black hole [79].

It turns out that the condition γ = −Qφ/2 reduces the problem of finding marginal

perturbations (4.1) to that of Ramond ground states in the theory M. In fact when we

take account of the N = 1 Liouville degrees of freedom and set γ = −Qφ/2, the Ramond

sector operator is given by

O(R) = σφe−Qφ2φO(R)M (4.6)

where σφ denotes the spin field associated with the Liouville fermion. If we recall the

criticality condition (2.2)

32(d− 2) + 32 + 3Q2φ + cM = 12 , (4.7)

and divide the formula by 24, we find

116 +Q2φ8 + cM24 = 10− d16. (4.8)

A state with h = (10−d)/16 is in fact the Ramond ground state for the internal space with

10− d dimensions. Thus the Ramond ground state of the system (N = 1 Liouville)×Mis constructed from the Ramond ground state h(O(R)

M ) = cM/24 of the theory M.

We thus have an one-to-one correspondence between the marginal perturbation and

Ramond ground state as

O(NS) = e−Qφ2φO(NS)M ⇐⇒ O(R) = σφe

−Qφ2φO(R)M (4.9)

Here O(NS)M and O(R)

M are related as (4.4). If one uses (4.3), one finds h(O(NS)) = 1/2.

Such a correspondence between Ramond ground states and marginal operators was first

pointed out in [47]. We will show below that in fact these operators O(NS) in the NS sector

are marginal perturbations preserving the special holonomies. We also note that O(NS)M is

the most relevant primary field since the Liouville exponential e−Qφ2φ has the maximum

dimension.

Our remaining task is to confirm that the cosmological constant operators defined here

are really marginal deformations preserving the spacetime SUSY. To this aim let us recall

40

the discussions given in [47]. First of all, the SCFT characterizing G2 holonomy contains

the tri-critical Ising model, and the energy momentum tensor is decomposed as

T = T tri + T r, T tri(z)T r(w) ∼ 0 ,

where T tri and T r satisfy the Virasoro algebra with central charge 7/10 and 49/5. We

express the conformal weights as (htri, hr) for T tri and T r respectively. We only treat here

operators with the same right moving quantum numbers with the left moving ones, and

focus only on the left movers.

As shown in [47], the deformation (4.1) preserves the spacetime supersymmetry and

exactly marginal, if and only if e−Qφ2

φO(NS)M is a primary of the type (htri, hr) =

(110, 2

5

).

Its corresponding Ramond sector operator

σφ±e

−Qφ2

φO(R)M , (4.10)

(generated by the “spectral flow operator”(

716, 0

)) must then be an operator of the type(

380, 2

5

). The operator (4.10) is doubly degenerate because of the spin field σφ

±, and we

must take a suitable linear combination compatible with the GSO projection, namely, the

mutual locality with spacetime supercharges.

In the Spin(7) holonomy cases almost the same argument works. The energy momen-

tum tensor can be decomposed into the Ising part and the rest, and we write the conformal

weights as (hIsi, hr). If and only if e−Qφ2

φO(NS)M is a primary of the type (hIsi, hr) =

(116, 7

16

),

the deformation (4.1) is an exactly marginal operator preserving Spin(7)-holonomy. More-

over, such an NS primary corresponds to a Ramond state of the dimension(

116, 7

16

).

Now, let us confirm that these are indeed the cases for our cosmological constant

operators (4.1). In the G2 holonomy case, the Ramond ground state operator must be

either(

380, 2

5

)or

(716, 0

). In our construction T tri is made up of sectors U(1)3/2 × ψφ and

does not contain the Liouville field φ. We can hence conclude that hr 6= 0 for the operator

(4.10). Consequently the operator (4.10) must be of the(

380, 2

5

)type.

In the Spin(7) holonomy case, the problem is a little more subtle. There are three

types of Ramond ground states:(0, 1

2

),(

12, 0

),(

116, 7

16

). Since hr 6= 0 holds by the same

reason as G2 holonomy case, the remaining possibilities are(0, 1

2

)or

(116, 7

16

). Actually,

the doubly degenerate operators (4.10) can be either of these types. However, it is possible

to show that the operator(

116, 7

16

), which has a mutually non-local OPE with the spectral

flow operator(

12, 0

), survives after the GSO projection.

In conclusion, the problem of enumerating marginal perturbations is reduced to the

classification of Ramond ground states of the conformal theoryM in the case of holonomies

G2 and Spin(7). It turns out that this also amounts to enumerating conformal blocks

F(∗)2 (h = 1/2; τ) defined in appendix B, appearing in the modular invariant partition

41

functions. Each function in the NS sector F(NS)2 (h = 1/2; τ) is one-to-one correspondence

with the operator e−Qφ2

φO(NS)M considered above.

We finally make a comment on the vacua with N = 2 worldsheet SUSY;

Rφ ×M ∼= (Rφ × S1Y )×M/U(1) , (4.11)

where M/U(1) is assumed to be an N = 2 SCFT and Rφ × S1Y denotes the N = 2

Liouville theory. It is not difficult to show that the cosmological constant operator (4.1)

does not preserve the worldsheet N = 2 SUSY. The easiest way to see it is to observe

that the integrality of the U(1)R-charge fails for the operator of the type (4.1). Thus we

now need to shift the exponent γ away from −Qφ/2. Typical operators which preserve

N = 2 SUSY are now written as ;[G−− 1

2

,[G−− 1

2

,O(NS)M/U(1) exp γ (φ+ iY )

]]+ (c.c.) ,

− Qφ

2γ + h(O(NS)

M/U(1)) =1

2, (4.12)

where O(NS)M/U(1) denotes a (cc) chiral primary field in the M/U(1) sector. Again for the

cases γ > −Qφ/2, (4.12) is non-normalizable and corresponds to a coupling constant, while

the operators γ < −Qφ/2 describe the normalizable moduli that resolve the singularity.

The most relevant primary is of course O(NS)M/U(1) = id., which corresponds to γ = −1/Qφ

and the well-known Liouville potential for the N = 2 Liouville theory. In the case of

singular Calabi-Yau n-folds, all of these operators (4.12) are normalizable for n = 2 while

non-normalizable for n = 4. In the CY3 case, one “half” of them are normalizable and

the remaining are non-normalizable. See for the detail [39, 29, 40, 31, 71].

4.2 Ramond Ground States in the N = 1 Supercoset CFT’s

As we have shown, finding the possible marginal deformations preserving Spin(7) or G2

holonomies amounts to the classification of Ramond ground states in the supercoset theory

M. We start by summarizing how to analyse this problem. It is quite reminiscent of the

analysis on the chiral rings in the Kazama-Suzuki models [76].

Let us recall the structure of N = 1 coset (3.1) and again assume H = H0 × H1 ×· · ·×Hr, where H0 denotes the abelian part and Hi are simple parts. For each Hi (i 6= 0),

the conformal dimension of the highest weight state λ(i) is evaluated as

h(λ(i)) =(λ(i), λ(i) + 2ρ(i))i

2(ki + h∗i )=|λ(i) + ρ(i)|2i2(ki + h∗i )

+cHi

24− dimHi

24

=|λ(i) + ρ(i)|22(k + g∗)

+cHi

24− dimHi

24, (4.13)

42

where ρ(i) denotes the Weyl vector of H(i) and ki is defined in (3.3). The norm | |i is

associated with the inner product ( , )i. We here made use of the well-known Freudenthal-

de Vries strange formula: (ρ(i), ρ(i))i = h∗i dimHi/12, and cHi≡ ki dimHi

ki+h∗idenotes the central

charge of (Hi)ki. This formula also applies to the abelian part if we set ρ(0) = 0. We thus

find that the conformal dimension of the highest weight state (Λ, s = 1, {λ(i)}) of the

Ramond sector becomes

h(Λ, s = 1, {λ(i)}) =(Λ,Λ + 2ρG)

2(k + g∗)+D

16−

r∑i=0

(λ(i), λ(i) + 2ρ(i))i

2(ki + h∗i )(4.14)

=|Λ + ρG|22(k + g∗)

−r∑

i=0

|λ(i) + ρ(i)|22(k + g∗)

+cM24

, (4.15)

where ρG is the Weyl vector of G, D ≡ dimG/H, and cM is given in (3.4). This relation is

valid only when the representation {λ(i)} is included in the representation Λ×((co)spinor)

as embedding of finite dimensional Lie algebra G × SO(D) ⊃ H . Therefore we find the

relation for the Ramond ground state

|Λ + ρG|2 − |λ+ ρH |2 = 0, (4.16)

In this equation, we define λ =∑

i λ(i) and ρH =

∑i ρ

(i). Formula (4.16) has been known

in N = 2 coset theories [76] and is now generalized to the case of N = 1 coset theories

with spacetime supersymmetry.

Note that the condition (4.16) does not depend on the level k of the affine algebra.

The level k enters only through the restriction on the possible set of representations Λ.

Because of the unitarity, the Λ =∑

i Λiωi must satisfy the following relations.

Λi ∈ Z, Λi ≥ 0, (θ,Λ) ≤ k. (4.17)

Only the finite number of Λ’s satisfy these relations.

In the next subsection we explicitly analyse the spectra of Ramond ground states. We

treat all the cases of simple group G, and also the example M = SU(2)3/SU(2) of G2

holonomy (3.47).

4.3 Spin(7) Holonomy Cases

We first study the Ramond ground states in the Spin(7) holonomy cases. The wanted

states are labeled by (Λ, s = 1, λ), where Λ is the integrable highest weight of Gk and λ is

the integrable highest weight of H . We also use the label Λ =∑

i Λiωi and λ =∑

i λiω′i,

where ωi and ω′i are the fundamental weights of G and H respectively.

43

1. SO(7)/G2: The relevant supercoset is

M =SO(7)k × SO(7)1

(G2)k+1(4.18)

Let us denote the highest weight of SO(7)k, SO(7)1, (G2)k+1 by Λ =∑3

j=1 Λjωj, s =

0, 1, 2,−1, λ =∑3

j=1 λjω′j, respectively. In this case, the Ramond ground states

satisfy

Λ3 = 2Λ1 + 1, λ1 = Λ2, λ2 = Λ1 + Λ3 + 1 (4.19)

in addition to the relations (4.17). The simplest example of the Ramond ground

states is Λ1 = 1,Λ2 = Λ3 = 0, which means Λ is the highest weight of spinor

representation of SO(7) and λ is the highest weight of 27 dimensional representation

of G2.

2. SU(3)/U(1): The relevant supercoset is written as

M =SU(3)k × SO(7)1

U(1), (4.20)

and the level of U(1) is determined when we fix the embedding. Consider the

N(m, `) type coset where U(1) lies along the direction ν = (m− `)ω1 + (m+ 2`)ω2.

We set m and ` relatively prime, and (m − `) ≥ 0, (m + 2`) ≥ 0. If we use

a canonically normalized U(1) charge p (which means that the dimension of an

exponential operator becomes p2/2), the Ramond ground state is obtained if it

satisfies either of the following two conditions

(a) (m− `)(Λ2 + 1) = (m+ 2`)(Λ1 + 1), p =m(Λ1 + 1) + (m+ `)(Λ2 + 1)√

2(k + 3)(m2 + `2 −m`).

(4.21)

(b) (m− `)(Λ1 + 1) = (m+ 2`)(Λ2 + 1), p = −m(Λ2 + 1) + (m+ `)(Λ1 + 1)√2(k + 3)(m2 + `2 −m`)

.

(4.22)

As we already mentioned, we have the N = 4 enhanced SUSY for the special case

of m = ` = 1, and the N = 1 cosmological terms (4.1) cannot be applied in this

case. We would like to further discuss this point elsewhere.

3. SO(5)/SO(3)max : The relevant supercoset is expressed as (3.60);

M =SO(5)k × SO(7)1

SU(2)10k+28

(4.23)

44

If we denote by (`+1) the the dimension of the representation of SU(2), the Ramond

ground state satisfies the relation

Λ2 = 2Λ1 + 1, ` = 2Λ1 + 5Λ2 + 6. (4.24)

The number of the Ramond ground state is evaluated as[k + 1

3

],

where [∗] denotes the Gauss symbol.

The simplest example of the Ramond ground state is Λ1 = 0,Λ2 = 1, which means

Λ is the highest weight of spinor representation of SO(5) and λ is the highest weight

of 12 dimensional representation of SU(2).

4.4 G2 Holonomy Cases

We next analyse the Ramond ground states for the G2 holonomy case. In these cases, the

coset fermions SO(6)1 has two representations in Ramond sector: spinor and cospinor.

We express these representation by s = ±1 respectively.

1. G2/SU(3): The relevant supercoset is given as

M =(G2)k × SO(6)1

SU(3)k+1. (4.25)

A Ramond ground state is obtained if either one of the following conditions is

satisfied

(a) s = −1, λ1 = Λ1, λ2 = Λ1 + Λ2 + 1. (4.26)

(b) s = 1, λ2 = Λ1, λ1 = Λ1 + Λ2 + 1. (4.27)

The number of the Ramond ground states is evaluated as follows;

1

2(k + 2)(k + 3)−

[k + 2

2

]. (4.28)

The simplest Ramond ground states come from the basic representation of G2, and

represented as (Λ = 0, s = −1, λ1 = 0, λ2 = 1) and (Λ = 0, s = +1, λ1 = 1, λ2 = 0).

45

2. SO(5)/(SU(2) × U(1)): The relevant coset corresponds to the (non-HSS) CP3

(3.32);

M =SO(5)k × SO(6)1

SU(2)k+1 × U(1)k+3

. (4.29)

We denote by (` + 1) the dimension of the representation of SU(2), and by m the

charge of U(1) normalized so that the conformal dimension of its vertex operator is

given by m2

4(k+3). In this notation, a Ramond ground state is obtained if it satisfies

one of the following four conditions

(a) s = −1, ` = Λ1, m = Λ1 + Λ2 + 2. (4.30)

(b) s = +1, ` = Λ1 + Λ2 + 1, m = Λ1 + 1 (4.31)

(c) s = −1, ` = Λ1 + Λ2 + 1, m = −Λ1 − 1 (4.32)

(d) s = +1, ` = Λ1, m = −Λ1 − Λ2 − 2 (4.33)

As a result, there are four Ramond ground states for each representation Λ of SO(5).

However there are the field identifications originating from the outer automorphism

Z2;

(Λ1,Λ2, s, `,m) ∼= (k − Λ1 − Λ2,Λ2, s+ 2, k − `,m+ (k + 3)) . (4.34)

Hence, the number of Ramond ground states is given by

(k + 1)(k + 2). (4.35)

Simplest ground states are comes from the basic representation of SO(5): (Λ =

0, s = 1, ` = 1, m = 1) and (Λ = 0, s = 1, ` = 0, m = −2).

3. SU(3)/U(1)2 : The relevant coset is characterized by the charge lattice Γ of U(1)2.

We here only consider the simplest case Γ = Zν1 + Zν2, where ν1 and ν2 are defined

in (3.35), leading to

M =SU(3)k × SO(6)1

U(1)k+3 × U(1)3(k+3)

. (4.36)

We label the U(1) charge by λ = λ1ω1 + λ2ω2 normalized so that the dimension

of the vertex operator is given by (λ,λ)2(k+3)

. The Ramond ground states satisfy the

condition

w(Λ + ρ) = λ, s = −ε(w), (4.37)

46

for an element w of the SU(3) Weyl group, where ε(w) is the signature of w. We here

note a slightly non-trivial point. The charge lattice Γ here is not invariant under the

outer automorphism Z3 which usually implies the necessity of field identification.

We thus should consider the spectrum without field identification contrary to the

standard Kazama-Suzuki coset Γ =√k + 3Q. Consequently, the number of the

Ramond ground states becomes

3(k + 1)(k + 2). (4.38)

The simplest Ramond ground states are expressed as (Λ = 0, s = −1, λ1 = 1, λ2 = 1)

and its Weyl transforms.

4. SU(2)3/SU(2): The supercoset is expressed as (3.47);

M =SU(2)k1 × SU(2)k2 × SU(2)k3 × SO(6)1

SU(2)k1+k2+k3+4. (4.39)

Let us denote the dimension of representation by `j +1, (j = 1, 2, 3) for each SU(2)

factor in the numerator, and the one in the denominator by `4 + 1 One obtains a

Ramond ground state if

`1 + 1

k1 + 2=`2 + 1

k2 + 2=`3 + 1

k3 + 2,

`4 = `1 + `2 + `3 + 3, s = ±1. (4.40)

are satisfied. In this case, by denoting the greatest common divisor of {kj + 2} as

p, the number of Ramond ground states becomes equal to (p− 1), when we take a

suitable field identification into account.

5 Discussions

In this paper we have investigated aspects of superstring vacua of the type:

Rd−1,1 × (N = 1 Liouville)× (N = 1 supercoset CFT on G/H) ,

motivated by an analogy with the geometrical cone constructions of special holonomy

manifolds over the Einstein homogeneous spaces G/H. We made an almost exhaustive

analysis and obtained results which led in most cases to the same amount of supersym-

metries as expected from the geometrical approach.

While these results seem satisfactory, we should also emphasize the obvious difference

between our and geometrical approaches. In the non-linear σ-model on the geometrical

47

cone over G/H, the physics depends on the choice of its metric, and possibly on the global

topology. On the other hand, in our algebraic approach based on the coset CFT’s, the

vacua are defined associated with the affine Lie algebras for G, H and further with the

choice of embedding of Lie (H) into Lie (G). We do not use information on the metric

structure and global topology of the manifold G/H explicitly. (It will be interesting

to check if one obtains metrics used in the geometrical cone constructions when one

computes the natural metric on homogeneous space making use of our embedding of H

into G described in various examples). Actually we often encountered examples where the

same manifold has several different coset realizations. They are of course equivalent in

the sense of non-linear σ-model, but not necessarily equivalent as the coset CFT’s. The

typical example is the case of round sphere. The geometrical cone over a round sphere is

a trivial flat space and leads to a vacua with a maximal amount of SUSY, while we have

several inequivalent coset CFT’s based on a round sphere possessing different amount of

supercharges.

Important questions for our analysis may as follows:

1. How can we identify our string vacua with the known solutions in supergravity

theories?

2. How can we attach a physical meaning to the levels of current algebras in our

construction?

3. What is the precise geometrical interpretation of the Ramond ground states dis-

cussed in section 4?

These problems are deeply related with each other and seem difficult in particular in the

cases of G2 and Spin(7) holonomies.

However, we now would like to point out some suggestive examples: For the d = 4

vacua with M = (SU(2)k1 × SU(2)k2)/U(1), we observed that the spacetime SUSY is

allowed only for the case of U(1) embedding p = q in (3.21) which corresponds to the

Einstein homogeneous space T 1,1. Moreover, it is easy to see that the simplest case

k1 = k2 = 0 is equivalent to the string theory on conifold as discussed in [26]. The brane

construction of conifold was explored in [80] based on the ALE fibration and the well-

known ALE-NS5 correspondence [27, 68]. It is realized as a system of two intersecting

NS5-branes, both of which fill the 4-dimensional Minkowski spacetime. Because the level

of WZW-model is translated into the brane charge of NS5, it is natural to suppose that

the higher level models M = (SU(2)k1 × SU(2)k2)/U(1) could be associated with a

configuration of intersecting stacks of NS5-branes with brane charges k1 and k2. In fact,

it is known [29, 33] that the vacua with k1 = k, k2 = 0 are understood as the ADE

hierarchy of rational singularities, or the wrapped NS5 branes with charge ' k.

48

On the other hand, the brane interpretation for the known supergravity solutions of

G2 (and Spin(7)) holonomy have been discussed in [5, 81]. In [5] the relevant brane

configurations are identified as D6-brane wrapped around special Lagrangian cycles (“L-

pictures”), which are again reinterpreted as the intersecting NS5-branes by some duality

web [81]. The cases of intersecting stacks of NS5-branes still describe the vacua with the

correct number of spacetime SUSY’s, although the explicit supergravity solutions are not

available. It seems again plausible to assume that the charges of stacked NS5-branes are

incorporated as the levels of WZW models in our construction.

Encouraged by these observations, we conjecture for the string vacua with G2 and

Spin(7) holonomies;

1. Our coset CFT models for the nearly Kahler (weak G2) homogeneous spaces Gk/H∗should be identified as the geometrical G2 (Spin(7)) cones over them in the special

case of zero level k = 0.

2. The non-zero level models could be associated with the same configurations of in-

tersecting NS5-branes as for the geometrical G2 (Spin(7)) cones considered above,

with the suitable numbers of stacked NS5-branes.

3. Marginal deformations discussed in section 4 correspond to various motions of in-

tersecting NS5-branes preserving the spacetime SUSY.

In this paper we have assumed the special value of the Liouville momentum γ = −Qφ/2

and have constructed associated marginal perturbation operators. An obvious question is

if one can construct other marginal operators with a different choice for the momentum.

It seems that this is a problem whose answer depends to some extent on how one defines

the Liouville theory. As is well-known there are two branches for the Liouville momentum

(i) “discrete series” : γ ∈ R (5.1)

(ii) “principal series” : γ = −Qφ2 + ip, p ∈ R (5.2)

and our choice −Qφ/2 was special in the sense that with this value of γ the dimension

of the Liouville exponential eγφ is maximum for the series (i) and minimum for the series

(ii). Relevant range of the “discrete series” here is γ < −Qφ/2 which provides operators

peaked around the tip of the cone and possibly resolve its singular behavior.

When one takes the Liouville momentum from the “discrete series” γ < −Qφ/2,

the dimension h(eγφ) could become negative and large as −γ becomes large. One may

possibly construct marginal operators by pairing such an exponential with a field from

the matter sector M with a large positive dimension. However, such a possibility seems

rather unphysical since the Liouville exponential with a large negative γ should effect

49

strongly the physics around φ ≈ −∞ while matter fields of high dimensions should be

largely irrelevant.

Unitarity of the Liouville theory is a difficult and subtle problem, however, it seems

generally agreed that the theory becomes unitary when one restricts oneself to the sector

of “principal series”. If one takes this point of view, the value γ = −Qφ/2 appears to be

the unique choice for marginal operators since this is the value for which the operator eγφ

is real and has the lowest dimension. In fact the central charge of the N = 1 Liouville

system is given by cL = 3/2 + 3Q2φ and the operator σφe−Qφ2φ saturates its BPS bound

cL/24 = 1/16 +Q2φ/8.

Thus we further conjecture

4. Marginal operators we have constructed exhaust all possible marginal perturbations

of the N = 1 coset model of string vacua for manifolds with special holonomy.

In order to discuss these conjectures it is quite important to establish the dictionary

translating the cosmological constant operators into the geometrical data. To this aim, a

possible future direction may be the boundary state analysis along the line similar to [71].

It may be also interesting and challenging to try to generalize the concept of chiral rings

characteristic for the N = 2 string vacua [76]. It is quite suggestive that our cosmological

constant operators are defined in one-to-one correspondence with the Ramond ground

states for the “base” M, implying some cohomological structure behind the system. The

approach based on the real Landau-Ginzburg theory may also be significant.

Acknowledgements

We would like to thank G. M. Sotkov for informing us the work [55] after our previous

paper [45] was published. We also would like to thank H. Kanno for useful information

on special holonomy cones.

Research of T.E. and Y.S. is supported in part by a Grant-in-Aid from Japan Ministry

of Education, Culture, Sports, Science and Technology. Research of S.Y. is supported in

part by Soryushi Shogakukai.

50

Appendix A : Notations

Set q := e2πiτ , y := e2πiz;

θ1(τ, z) = i∞∑

n=−∞(−1)nq(n−1/2)2/2yn−1/2 ≡ 2 sin(πz)q1/8

∞∏m=1

(1− qm)(1− yqm)(1− y−1qm),

θ2(τ, z) =

∞∑n=−∞

q(n−1/2)2/2yn−1/2 ≡ 2 cos(πz)q1/8

∞∏m=1

(1− qm)(1 + yqm)(1 + y−1qm),

θ3(τ, z) =∞∑

n=−∞qn2/2yn ≡

∞∏m=1

(1− qm)(1 + yqm−1/2)(1 + y−1qm−1/2),

θ4(τ, z) =

∞∑n=−∞

(−1)nqn2/2yn ≡∞∏

m=1

(1− qm)(1− yqm−1/2)(1− y−1qm−1/2).

(A.1)

Θm,k(τ, z) =

∞∑n=−∞

qk(n+ m2k

)2yk(n+ m2k

), (A.2)

Θm,k(τ, z) =∞∑

n=−∞(−1)nqk(n+ m

2k)2yk(n+ m

2k). (A.3)

We often use the abbreviations; θi ≡ θi(τ, 0) (θ1 ≡ 0), Θm,k(τ) ≡ Θm,k(τ, 0), Θm,k(τ) ≡Θm,k(τ, 0). We also set

η(τ) = q1/24∞∏

n=1

(1− qn). (A.4)

The product formula of theta functions is useful for our analysis;

Θm,k(τ, z)Θm′,k′(τ, z′) =

∑r∈Zk+k′

Θmk′−m′k+2kk′r,kk′(k+k′)(τ, u)Θm+m′+2kr,k+k′(τ, v), (A.5)

where we set u =z − z′

k + k′, v =

kz + k′z′

k + k′.

The next formula is also useful

Θm/p,k/p(τ, z) = Θm,k(τ/p, z/p) =∑r∈Zp

Θm+2kr,pk(τ, z/p) , (A.6)

where m, k are some real numbers and p is an integer.

51

Appendix B : Massive Characters of the Extended

Chiral Algebras Associated to Special Holonomies

Here we summarize the massive character formulas of the extended chiral algebras

characterizing the special holonomy manifolds. For simplicity we shall only focus on the

NS sector. The characters for other spin structures are obtained by making the half-

integral spectral flows as follows;

Ch(NS)(∗; τ, z) = Ch(NS)(∗; τ, z +1

2)

Ch(R)(∗; τ, z) = qc24 y

c6 Ch(NS)(∗; τ, z +

τ

2)

Ch(R)(∗; τ, z) = qc24 y

c6 Ch(NS)(∗; τ, z +

τ

2+

1

2) . (B.1)

1. Sp(k)-holonomy (k = 1, 2) :

This case corresponds to hyper Kahler manifolds of the real dimension 4k. The relevant

chiral algebra is the (small) N = 4 superconformal algebra with the level k (c = 6k), as

is well-known. The massive representations are labeled by the conformal weight h and

the SU(2) spin `/2, and the unitarity requires the constraints h ≥ `/2. The character

formulas are given in [59];

Ch(NS)(h, `; τ, z) =qh− (`+1)2

4(k+1)− 1

4

η(τ)

(θ3(τ, z)

η(τ)

)2

χ(k−1)` (τ, z) , (0 ≤ ` ≤ k − 1) , (B.2)

where χ(k)` (τ, z) denotes the character of SU(2)k with the spin `/2 (0 ≤ ` ≤ k) represen-

tation;

χ(k)` (τ, z) =

Θ`+1,k+2 −Θ−`−1,k+2

Θ1,2 −Θ−1,2

(τ, z) . (B.3)

We note that

χ(1)0 (τ, z) =

Θ0,1(τ, 2z)

η(τ)

χ(1)1 (τ, z) =

Θ1,1(τ, 2z)

η(τ). (B.4)

2. SU(n)-holonomy :

This case corresponds to the Calabi-Yau n-fold (CYn) compactification. The extended

chiral algebras are defined by adding the integral spectral flow operator, which corresponds

to the holomorphic n-form in CYn, to the N = 2 superconformal algebra with c = 3n.

52

These conformal algebras are not Lie algebras but are W-algebras except for the SU(2)-

holonomy case, which of course reduces to the above Sp(1) case. For the SU(3)-holonomy

the spectral flow operator generates the subsector described by SO(6)1/SU(3)1∼= U(1)3/2,

which is equivalent with the N = 2 minimal model of level 1. The corresponding sub-

sector for the SU(4)-holonomy is SO(8)1/SU(4)1∼= U(1)2, which is equivalent with the

conformal system of a complex fermion.

The massive representations are labeled by the conformal weight h and the U(1)R

charge Q with the unitarity condition h ≥ Q/2. The character formulas for SU(3)-

holonomy are given in [60], and those for SU(4)-holonomy are given in [62].

• SU(3)-holonomy : We have two continuous series;

Ch(NS)(h,Q = 0; τ, z) =qh− 1

4

η(τ)

θ3(τ, z)

η(τ)

Θ0,1(τ, 2z)

η(τ),

Ch(NS)(h, |Q| = 1; τ, z) =qh− 1

2

η(τ)

θ3(τ, z)

η(τ)

Θ1,1(τ, 2z)

η(τ). (B.5)

The vacuum state is doubly degenerate for the second case (Q = 1 and Q = −1).

• SU(4)-holonomy : We have three continuous series;

Ch(NS)(h,Q = 0; τ, z) =qh− 3

8

η(τ)

θ3(τ, z)

η(τ)

Θ0, 32(τ, 2z)

η(τ),

Ch(NS)(h,Q = ±1; τ, z) =qh− 13

24

η(τ)

θ3(τ, z)

η(τ)

Θ±1, 32(τ, 2z)

η(τ). (B.6)

3. G2 and Spin(7)-holonomies :

The chiral algebras associated to the G2 and Spin(7)-holonomies are again the W-

algebra like extensions of N = 1 superconformal algebra explicitly defined in [47]10. A

characteristic feature of the G2 extended algebra is the existence of the tri-critical Ising

model (∼= SO(7)1/(G2)1), and that for the Spin(7) extended algebra is the Ising model

(∼= SO(8)1/SO(7)1) as discussed in [47]. The unitary massive representations for these

algebras are classified in [49, 50]: two continuous series with h ≥ 0 and h ≥ 1/2 exist

for the each case. Unfortunately, the character formulas for these representations have

not been worked out. Nevertheless, based on the existence of spacetime SUSY as well as

the worldsheet N = 1 superconformal symmetry, it seems plausible to propose that the

conformal blocks corresponding to the massive representations should be expanded into

the following functions:10In some literature the G2 extended algebra is denoted as SW(3/2, 3/2, 2)-algebra (c = 21/2) and the

Spin(7) extended algebra is done as SW(3/2, 2)-algebra (c = 12).

53

• Spin(7)-holonomy : For the NS sector,

F(NS)1 (h; τ) =

qh− 49120

η(τ)

√θ3η

(τ)χtri, (NS)0 (τ)

≡ qh− 49120

η(τ)

(χIsing

0 (τ)χtri0 (τ) + χIsing

1/2 (τ)χtri3/2(τ) + χIsing

1/16 (τ)χtri7/16(τ)

),

(h ≥ 0)

F(NS)2 (h; τ) =

qh− 110− 49

120

η(τ)

√θ3η

(τ)χtri, (NS)1/10 (τ)

≡ qh− 110− 49

120

η(τ)

(χIsing

0 (τ)χtri3/5(τ) + χIsing

1/2 (τ)χtri1/10(τ) + χIsing

1/16 (τ)χtri3/80(τ)

),

(h ≥ 1/2) (B.7)

where we have written χtri, (NS)h , χtri

h for the N = 1 and N = 0 characters of the

tri-critical Ising model, and χIsingh for the Ising model. The second lines are con-

sistent with the decompositions of massive representations of the Spin(7) extended

algebra with respect to the Virasoro modules of Ising model presented in [49]. This

fact suggests that the functions F(NS)i (h; τ) may in fact be the massive characters

although we have not been able to prove this.

• G2-holonomy : For the NS sector, the wanted functions are given as

F(NS)1 (h; τ) =

qh− 13

η(τ)

√θ3η

(τ)Θ0,3/2

η(τ)

≡ qh− 13

η(τ)

(χ

tri, (NS)0 χPotts

0 + χtri, (NS)1/10 χPotts

2/5

)(τ) , (h ≥ 0)

F(NS)2 (h; τ) =

qh− 12

η(τ)

√θ3η

(τ)Θ1,3/2

η(τ)

≡ qh− 12

η(τ)

(χ

tri, (NS)0 χPotts

2/3 + χtri, (NS)1/10 χPotts

1/15

)(τ) , (h ≥ 1/2) (B.8)

where χtri, (NS)h again means the N = 1 character of tri-critical Ising and χPotts

h

denotes the character of the 3-state Potts model (Z3 Parafermion) that is defined

as the coset CFT SU(2)3/U(1)3. The second lines are easily derived using the

54

equivalence

SO(1)1 × SO(6)1

SU(3)1

∼= SO(7)1

(G2)1

× (G2)1

SU(3)1

∼= tri-critical Ising× 3-state Potts . (B.9)

They are consistent with the structures of unitary massive representations given in

[50].

55

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